Two quadratic equations can share a solution if one quadratic equation has a positive discriminant and another quadratic equation has a discriminant equal to 0
What is meant by quadratic equation?
A quadratic equation is a polynomial equation of degree 2, which means it involves an unknown variable (usually denoted as x) that is raised to the power of 2.
If one quadratic equation has a positive discriminant, it means that the quadratic equation has two distinct real roots. On the other hand, if another quadratic equation has a discriminant equal to 0, it means that the quadratic equation has only one real root that is repeated.
Therefore, it is possible for two quadratic equations to share a solution if one quadratic equation has a positive discriminant and another quadratic equation has a discriminant equal to 0, but only if the repeated root of the second quadratic equation is one of the two roots of the first quadratic equation.
To illustrate this, let's consider the following two quadratic equations:
Quadratic equation 1: [tex]$x^2 - 4x + 3 = 0$[/tex]
Quadratic equation 2: [tex]$x^2 - 2x + 1 = 0$[/tex]
The discriminant of quadratic equation 1 is [tex]$b^2-4ac = 4 - 4(1)(3) = -8$[/tex], which is negative. Therefore, quadratic equation 1 has two distinct real roots, which can be found using the quadratic formula:
[tex]$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$[/tex]
Substituting the values of a, b, and c from quadratic equation 1, we get:
[tex]$x=\frac{4\pm\sqrt{(-4)^2-4(1)(3)}}{2(1)}$[/tex]
[tex]$x=2\pm\sqrt{1}$[/tex]
[tex]x=3$ or $x=1$[/tex]
Therefore, the roots of quadratic equation 1 are [tex]x=3$ and $x=1$[/tex].
The discriminant of quadratic equation 2 is [tex]$b^2-4ac = 2^2 - 4(1)(1) = 0$[/tex]. Therefore, quadratic equation 2 has one repeated real root, which is:
[tex]$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$[/tex]
Substituting the values of a, b, and c from quadratic equation 2, we get:
[tex]$x=\frac{2\pm\sqrt{2^2-4(1)(1)}}{2(1)}$[/tex]
[tex]$x=1$[/tex] (repeated root)
As we can see, the repeated root of quadratic equation 2 is equal to one of the roots of quadratic equation 1, which is [tex]$x=1$[/tex]. Therefore, these two quadratic equations share a solution.
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Find the sum and simplify your answer completely.
3/4 + 7/12
Answer:
[tex]\frac{4}{3}[/tex]
Step-by-step explanation:
Please someone answer this for me
The scale drawing of the mural would be 594.36 cm wide and 1097.28 cm tall.
What are scale drawings?A scale drawing is a drawing that shows an object or area at a scale that differs from the real size of the object or area. A scale, which is a ratio comparing the size of the drawing to the size of the actual item or location, is used to make scale drawings. A scale of 1:50, for instance, indicates that one unit on the design corresponds to fifty in real life.
In several disciplines, including architecture, engineering, and design, scale drawings are employed in real-world situations. Scale drawings are used by architects to construct building floor plans, elevations, and sections. Scale drawings are used by engineers to develop mechanical systems and machinery.
Given that the dimensions of the mural is 13 feet by 24 feet.
Converting into cm we have:
width = 13 feet * 30.48 cm/foot = 396.24 cm
height = 24 feet * 30.48 cm/foot = 731.52 cm
Now using the scale we have:
3cm : 2ft = x cm : 396.24 cm
Using cross multiplication we have:
2ft * x cm = 3cm * 396.24 cm
x = (3cm * 396.24 cm) / 2ft = 594.36 cm
For the height, we have:
3cm : 2ft = y cm : 731.52 cm
2ft * y cm = 3cm * 731.52 cm
y = (3cm * 731.52 cm) / 2ft = 1097.28 cm
Hence, the scale drawing of the mural would be 594.36 cm wide and 1097.28 cm tall.
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A restaurant at the food court in a mall is offering a lunch special. The table shows the relationship between the number of side dishes and the total cost of the special.
Restaurant
Number of Side Dishes Total Cost
2 $8.50
4 $11.50
5 $13.00
8 $17.50
Which of the following graphs shows the relationship given in the table?
graph with the x axis labeled number of side dishes and the y axis labeled cost in dollars and a line going from the point 0 comma 7 and 25 hundredths through the point 3 comma 11 and 75 hundredths
graph with the x axis labeled number of side dishes and the y axis labeled cost in dollars and a line going from the point 0 comma 7 and 75 hundredths through the point 3 comma 12 and 25 hundredths
graph with the x axis labeled number of side dishes and the y axis labeled cost in dollars and a line going from the point 0 comma 5 and 25 hundredths through the point 3 comma 9 and 75 hundredths
graph with the x axis labeled number of side dishes and the y axis labeled cost in dollars and a line going from the point 0 comma 5 and 5 tenths through the point 5 comma 13
What angle ???? I’ll mark BRAINLIEST
Answer: ∠1 and ∠2 are supplementary angles.
Step-by-step explanation:
Complementary angles are two angles that add up to 90 degrees.
Supplementary angles are angles that add up to 180 degrees.
∠1 + ∠2 = 180 degrees, so it is a supplementary angle.
How do you find orthogonal projection on a line?
the orthogonal projection of a point P onto a line L can be expressed as: projection L(P) = ((P . u) ×u)·
Define orthogonal projectionOrthogonal projection is a mathematical operation that involves projecting a vector or point onto another vector or line in a way that the projection is perpendicular (or orthogonal) to the vector or line.
To find the orthogonal projection of a point on a line, you can follow these steps:Identify the point and the line: Let the point be P and the line be L.Find a vector that is parallel to the line: Let the vector be v.Find the unit vector that is parallel to the line: Normalize the vector v by dividing it by its magnitude to obtain the unit vector u.Find the vector from the origin to the point P: Let the vector be w.Calculate the dot product of w and u: This gives the magnitude of the projection of w onto u.Multiply u by the dot product obtained in step 5: This gives the projection vector.Add the projection vector to the origin: This gives the orthogonal projection of the point P onto the line L.In mathematical notation, the orthogonal projection of a point P onto a line L can be expressed as:
projection L(P) = ((P . u) × u)
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pls anwser need helpppp
A rocket is launched from a tower. The height of the rocket, y in feet, is related to the time after launch, x in seconds, by the given equation. Using this equation, find the maximum height reached by the rocket, to the nearest tenth of a foot. y=16x^2+173x+140
An equation is formed of two equal expressions. The maximum height reached by a rocket, to the nearest tenth of a foot is 1542.82 feet.
What is an equation?An equation is formed when two equal expressions are equated together with the help of an equal sign '='.
To find the maximum height through which the rocket will reach, we need to differentiate the given function, therefore, we can write,
y=-16x²+173x+140
dy/dx = -16(2x)+173
Substitute the value of dy/dx as 0, to get the value of x,
0 = -32x + 173
173 = 32x
x =5.406
Substitute the value of x in the equation to get the maximum height,
y=-16x²+173x+140
y=-16(5.406²) +173(5.406) +140
y= 467.59 + 935.23 + 140
= 1542.82 feet.
Hence, the maximum height reached by the rocket, to the nearest tenth of a foot is 1542.82 feet.
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What is y if
-1/2 = 3/8y
Answer: -1 1/3 (this is one and a third) or -4/3 (improper fraction)
Step-by-step explanation:
Solve for y by simplifying both sides of the equation then isolate the variable (y).
y = –5x + 10
y = –3x + 2
Answer:
Step-by-step explanation:
y = –5x + 10
y = –3x + 2
–5x + 10= –3x + 2
2x= 8
x=4
y= -5(4)+10
y= -20+10
y=(-10)
Answer:
Step-by-step explanation:
[tex]-5x + 10= -3x + 2[/tex] (since both equal y)
[tex]10=2x+2[/tex] ([tex]+5x[/tex] both sides)
[tex]8=2x[/tex] (-2 both sides)
[tex]x=4[/tex] (÷2 both sides)
Sub [tex]x=4[/tex] to find y:
[tex]y=-5(4)+10=-10[/tex]
Solution: x=4, y= -10
the area of a rectangle is 1,056 square inches. its length is 4 inches longer than 2 times its width. which equation can you use to find the width of the rectangle, w?
Answer:
The equation for the width is:
1056 = (2w+4) * w
And the solution is:
Width: 22 inches
Length: 48 inches
Step-by-step explanation:
Area of a rectangle = length * width
1056 = h * w Eq. 1
h = 2w + 4 Eq. 2
h = length
w = width
Replacing Eq. 2 in Eq. 1:
1056 = (2w+4) * w
1056 = 2w*w + 4*w
1056 = 2w² + 4w
2w² + 4w - 1056 = 0
w = {-4±√((4²)-(4*2*-1056))} / (2*2)
w = {-4±√(16 + 8448)} / 4
w = {-4±√8464} / 4
w = {-4±92} / 4
Is a geometric figure, therefore, only the positive value will be obtained
w = {-4+92}/4
w = {88} / 4
w = 22
From Eq. 2:
h = 2w+4
h = 2*22+4
h = 44+4
h = 48
Check:
From Eq. 2
1056 = h * w
1056 = 48 * 22
what pattern of growth are we currently exhibiting? exponential growth intrinsic growth logistic growth none of the other answer options is correct. geometric growth
Geometric growth is a type of exponential growth where a quantity or value increases at a fixed percentage rate over a certain period of time.
Geometric growth is a type of exponential growth where a quantity or value increases at a fixed percentage rate over a certain period of time. This means that the growth rate is proportional to the current size or value of the quantity, resulting in a continuously increasing growth rate.
Geometric growth is commonly observed in natural and social systems, such as population growth, compound interest, and the spread of infectious diseases. It is represented mathematically by an exponential function, where the base of the exponent is greater than one, indicating an increasing rate of growth over time.
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I have solved the question in general, as the given question is incomplete.
The complete question is:
What is geometric growth ?
I need helppppppppp please help me
Answer:
12.9cm
Step-by-step explanation:
Trapeziums QRVW and QRSP are similar because they share the same side 5.5 cm and their angles are corresponding to each other.
We can express the ratios of sides in both trapeziums to find WV.
Since side 5.5 is the same in each trapezium,
[tex]\frac{5.5}{WV}=\frac{5.5}{12.9}[/tex]
Thus, WV = 12.9cm
Hope this helps:)
if the given triangles are similar find the missing length
The length of PQ in triangle PQR is 110.
What are similar triangles ?
Similar triangles are two triangles that have the same shape but may differ in size. In other words, their corresponding angles are equal, and their corresponding sides are in proportion. This means that if you were to resize one triangle by a constant factor, the resulting triangle would still be similar to the original triangle.
For example, if you take a triangle and multiply all its side lengths by a factor of 2, you will end up with a triangle that is twice as large but still similar to the original triangle.
According to the question:
Since the triangles XYZ and PQR are similar, their corresponding sides are in proportion. That is:
XY/PQ = XZ/PR = YZ/QR
We can use this property to find the length of the missing side QP.
First, let's find the ratio of corresponding sides:
XY/PQ = XZ/PR
48/PQ = 55/110
Simplifying this equation, we get:
PQ = 48*110/55 = 96
So, PQ is 96.
Alternatively, we could use the other pair of corresponding sides:
XZ/PR = YZ/QR
55/110 = 73/146
Simplifying this equation, we get:
QP = 55*146/73 = 110
So, QP is also 110.
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a certain dj takes requests for songs at a party. assume that there are 120 people at the party, each of whom makes exactly one request for a song. all of their requests are made independently. assume that each person asks for a pop song with probability 0.37, a rock song with probability 0.2, or a rap song with probability 0.43. what is the probability that 50 or more requests are made for pop songs?
the probability that 50 or more requests are made for pop songs is 0.0967, or about 9.67%.
Define binomial distributionThe binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success.
We can use the binomial distribution to model this situation, where the number of trials is 120 (the number of people at the party), and the probability of success (making a request for a pop song) is 0.37. Let X be the number of requests for pop songs. Then:
The mean of X is μ = np = 120 x 0.37 = 44.4
The standard deviation of X is σ = sqrt(np(1-p)) = sqrt(120 x 0.37 x 0.63) = 4.32
To find the probability that 50 or more requests are made for pop songs, we can use the normal approximation to the binomial distribution, which applies when np >= 10 and n(1-p) >= 10.
Let Z be a standard normal random variable, then:
P(X >= 50) = P((X - μ)/σ >= (50 - μ)/σ)
= P(Z >= (50 - 44.4)/4.32)
= P(Z >= 1.2963)
= 1 - P(Z < 1.2963)
= 1 - 0.9033
= 0.0967
Therefore, the probability that 50 or more requests are made for pop songs is approximately 0.0967, or about 9.67%.
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Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)( (ln 2n) / ( (ln 6n) )( (ln2n)/ (ln6n) ) when n approaches infinity
The sequence converges to ln 2 / ln 6 as n approaches infinity.
How to find limit?
To determine whether the sequence converges or diverges, we need to take the limit of the sequence as n approaches infinity.
a_n = (ln 2n) / (ln 6n)
We can simplify this expression by applying logarithm rules:
a_n = (ln 2n) / (ln 6n) = (ln 2 + ln n) / (ln 6 + ln n)
Now we can use the fact that for any constant a>1, ln n is dominated by ln a as n approaches infinity, i.e., ln n / ln a approaches infinity as n approaches infinity.
Thus, as n approaches infinity, we have:
a_n = (ln 2 + ln n) / (ln 6 + ln n) ≈ ln 2 / ln 6
Therefore, the sequence converges to ln 2 / ln 6 as n approaches infinity.
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Which value of x makes the expression 3√53x
equivalent to 21√53?
The value of x that makes the expression 3√(53x) equivalent to 21√53 is 49.
What does expression mean?In mathematics, an expression is a combination of numbers, variables, and operations, typically written using mathematical symbols and/or mathematical notation. An expression can be a single number, variable, or combination of the two, or it can be a more complex combination of numbers, variables, and operators such as addition, subtraction, multiplication, division, exponentiation, or root extraction. Expressions can be used to represent a wide variety of mathematical concepts and relationships, such as equations, inequalities, functions, and geometric figures. They are an important part of mathematical language and are used to communicate mathematical ideas and concepts.
What does equation mean?In mathematics, an equation is a statement that shows the equality of two expressions. It typically involves one or more variables and mathematical operations such as addition, subtraction, multiplication, division, exponentiation, or root extraction. The expressions on both sides of the equation are separated by an equal sign (=), indicating that they have the same value. The purpose of an equation is to find the value of the variable(s) that makes the equation true. Solving an equation involves manipulating the expressions on both sides of the equation to isolate the variable and determine its value. Equations are used in a wide variety of mathematical applications, such as solving problems in algebra, geometry, calculus, and physics.
According to the given informationWe can start by equating the expression 3√(53x) to 21√53 and then solving for x.
3√(53x) = 21√53
Dividing both sides by 3√53, we get:
√x = 7
Squaring both sides of the equation, we get:
x = 49
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you decide to go to a rent-to-own company and purchase a laptop. the cost is $20.95 per month for 5 years. how much did you pay total at the end of the 5 years for the laptop?
You paid the total amount of $1257 at the end of 5 years.
Let us assume that c represents the cost per month.
So, c = $20.95
and let t represents the total number of months.
We know that there are 12 months in a year.
so, the number of months in 5 years would be:
12 × 5 = 60
So, t = 60 months
Using unitary method the total amount paid at the end of five years :
s = c × t
s = 20.95 × 60
s = $1257
Therefore, the total amount is $1257
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Which Operation would you use to solve each equation? Match each equation to the correct category
Use Addition to Solve
Use Subtraction to Solve
a-6.8=9.3
7=m+1.2
4+ y = 13
2=1-1
Equation one can be solved by additive property two by subtractive property.
What is an equation?A mathematical statement known as an equation consists of two algebraic expressions separated by equal signs (=) on either side.
It demonstrates the equality of the relationship between the printed statements on the left and right.
Left side equals right side in all formulas.
To find the values of unknowable variables, which stand in for unknowable quantities, you can solve equations.
A statement is not an equation if it lacks the equals sign.
When two expressions have the same value, a mathematical statement known as an equation will include the symbol "equal to" between them.
7=m+1.2
m=7-1.2
6.8
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What is the 2-digit subtraction with regrouping worksheets?
Answer:
To subtract two-digit numbers, use column form. Subtract one column at a time, starting with the Ones column. We regroup, or borrow. To regroup, take 1 from the Tens column and add 10 to the Ones column.
Step-by-step explanation:
Give both values of x that satisfy the equation
3 + 6 = x
X
Give your answers as decimals to 3 s.f.
Answer:
x = 6.464 and x = -0.464
Step-by-step explanation:
3/x + 6 = x
3 + 6x/x = x
3 + 6x = x²
x² - 6x - 3 = 0
x = 6 ± √36 - 4x - 3/2 = 6 ± √36 + 12/2
x = 6 ± √48/2 = 6 ± 4√3/2 = 3 ± 2√3
So x = 3 + 2√3 and 3 - 2√3
x = 6.464 and x = -0.464
5 Divided 1/4 As a Area Model
(A area model please)
Answer: You can comment if this is wrong but I don't know what an area model is entirely I might just I don't remember
|\ |\
| \ | \
| \| \
|___\|___\
1/4 1/5
Step-by-step explanation:
1. Draw a rectangular area model with two sections.
2. Label the left section with 1/4 and the right section with 1/5.
3. Divide the left section into 4 equal parts by drawing lines horizontally and vertically
4. Divide the right section into 5 equal parts by drawing lines horizontally and vertically.
5. Count the number of parts in the left section (4 parts) and the number of parts in the right section (5 parts).
6. Multiply the parts in the left section by the parts in the right section, which gives 4 x 5 =20.
7. The answer is 20/4 = 5.
PLEASE HELPPPP GIVING BRAINLIEST IF CORRECT
Answer: B
Step-by-step explanation:
Find the length of the hypotenuse of a 45°-45°-90° triangle with a leg
length of 8√8 centimeters
Answer:
Therefore, the length of the hypotenuse of the 45°-45°-90° triangle is 32 centimeters.
Step-by-step explanation:
In a 45°-45°-90° triangle, the two legs are congruent, which means that each leg is equal to the length of the hypotenuse divided by √2.
Let h be the length of the hypotenuse. Then we have:
leg = h/√2
Given that one of the legs is 8√8 centimeters, we can substitute this value into the equation above to solve for h:
8√8 = h/√2
Multiplying both sides by √2, we get:
8√8 x √2 = h
Simplifying, we get:
h = 8√16
h = 8 x 4
h = 32
Therefore, the length of the hypotenuse of the 45°-45°-90° triangle is 32 centimeters.
A nutrition label shows that there are 161 calories in 28 grams of dry roasted cashews. You eat 9 cashews totaling 12 grams.
You consumed approximately 69 calories from the 9 cashews you ate.
What is statistics?
Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data.
A nutrition label typically provides information about the nutrient content of a food item per serving size. In this case, the label states that 28 grams of dry roasted cashews contain 161 calories.
To determine the number of calories in a smaller portion of cashews, we need to use a proportion or ratio.
We start by calculating the number of calories per gram of cashews by dividing the total number of calories by the total weight of cashews:
161 calories / 28 grams = 5.75 calories per gram of cashews
This means that each gram of cashews contains about 5.75 calories.
Next, we need to figure out how many grams of cashews you ate. The question states that you ate 9 cashews, which weighed a total of 12 grams.
To find out the number of calories you consumed, we simply need to multiply the weight of the cashews (12 grams) by the number of calories per gram (5.75):
5.75 calories per gram x 12 grams = 69 calories
Therefore, you consumed approximately 69 calories from the 9 cashews you ate.
Complete question is:
A nutrition label shows that there are 161 calories in 28 grams of dry roasted cashews. You eat 9 cashews totaling 12 grams. How many calories did you consume?
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a political candidate has asked his/her assistant to conduct a poll to determine the percentage of people in the community that supports him/her. if the candidate wants a 1% margin of error at a 95% confidence level, what size of sample is needed? be sure to round accordingly. the candidate would need to survey people in the community in order to be within a 1% margin of error at a 95% confidence level.
In order to be within a 1% margin of error at a 95% confidence level, the candidate would need to survey 9604 people in the community. The size of the sample is 9604.
We have a political candidate who wants a 1% margin of error at a 95% confidence level. To find the size of the sample, we use the formula given below;
n = [ Z (α/2) / E ]²
Where,
n = the sample size required
Z (α/2) = the value obtained from the standard normal distribution for the given level of confidence
E = the margin of error
Here, α = 1 - 0.95 = 0.05
Z (α/2) = Z (0.025), is the z-value corresponding to 0.025 in the normal distribution table. It is found to be 1.96.
Now, substituting the given values in the formula,
n = (1.96² * 0.5 * 0.5) / 0.01² ⇒ 9604
Hence, the sample size required for the poll is 9604.
However, this is not feasible, as it would be very expensive and time-consuming. Therefore, we will round the answer to the nearest whole number as we are looking for a practical solution. Thus, the candidate would need to survey 9604 people in the community (rounded to the nearest whole number).
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solve this equasion 8x+12
worth 89 pints!!
pls asnweer
due in 5 minss
Answer:
a. 4
b. -2
c. 0,5
it js wants u to say what the number line represents
Step-by-step explanation:
a quality control inspector has drawn a sample of 16 light bulbs from a recent production lot. suppose 20% of the bulbs in the lot are defective. what is the probability that between 6 and 9 (both inclusive) bulbs from the sample are defective? round your answer to four decimal places.
The probability that between 6 and 9 (both inclusive) bulbs from the sample are defective is 0.5362
This is a binomial distribution problem, where the probability of success (defective bulb) is 0.2 and the probability of failure (non-defective bulb) is 0.8. We need to find the probability that between 6 and 9 (both inclusive) bulbs out of 16 bulbs in the sample are defective.
We can use the binomial probability formula to solve this problem
P(6 ≤ X ≤ 9) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9)
where X is the number of defective bulbs in the sample.
P(X = k) = (n choose k) × p^k × (1-p)^(n-k)
where n is the sample size, k is the number of defective bulbs, p is the probability of a defective bulb, and (n choose k) is the binomial coefficient.
Using this formula, we can calculate the probability for each value of X and sum them up to get the probability for the range 6 to 9.
P(X = 6) = (16 choose 6) × 0.2^6 × 0.8^10 = 0.0881
P(X = 7) = (16 choose 7) × 0.2^7 × 0.8^9 = 0.1409
P(X = 8) = (16 choose 8) × 0.2^8 × 0.8^8 = 0.1606
P(X = 9) = (16 choose 9) × 0.2^9 × 0.8^7 = 0.1462
Therefore, the probability that between 6 and 9 (both inclusive) bulbs from the sample are defective is
P(6 ≤ X ≤ 9) = 0.0881 + 0.1409 + 0.1606 + 0.1462 = 0.5362
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Write the set notation to represent the shaded portion in the given Venn-diagram
The set notation that represents the shaded portion in the given Venn-diagram is: A ∪ B − (A ∩ B).
What is the set notation of the Venn Diagram?The union of the sets A and B simply denotes everything which is in either A or B, as represented by the shaded region in the given venn diagram. The intersection of two sets is that which is in both sets, as represented by the unshaded region in the following Venn diagram
The union of A and B is the entirety of all that is in either A or B, as represented by the shaded area inside the given venn diagram.
The intersection of sets is in each sets, as represented via way of means of the magenta shaded area withinside the following Venn diagram.
Thus, it is A ∪ B − (A ∩ B).
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20 Points:
How many ways are there to form a 5 person committee from 2 freshmen, 3 sophomores, 2 juniors, and 2 seniors if there cannot be the same number of juniors and seniors? Note that people are distinct.
=================================================
Explanation:
2 freshmen + 3 sophomores + 2 juniors + 2 seniors = 9 students total.
Let's consider the cases where we have the same number of juniors and seniors. We'll then take the complement of this to get the final answer.
Case A will look at having 1 of each junior and senior.Case B will look at having 2 each of juniors and seniors.-----------
Case A: There is 1 junior and 1 senior
There are 2 ways to pick a junior and 2 ways to pick a senior. That's 2*2 = 4 ways so far.
Then we have 9-4 = 5 students left to pick from (i.e. 2 freshmen+3 sophomores = 5 students left) and we have 5-2 = 3 seats to fill. Order doesn't matter on the committee since each person has equal rank.
We use the nCr combination formula.
n = 5 students
r = 3 seats to fill
n C r = (n!)/(r!(n-r)!)
5 C 3 = (5!)/(3!*(5-3)!)
5 C 3 = (5!)/(3!*2!)
5 C 3 = (5*4*3!)/(3!*2!)
5 C 3 = (5*4)/(2!)
5 C 3 = (5*4)/(2*1)
5 C 3 = (20)/(2)
5 C 3 = 10
There are 10 ways to fill the remaining 3 seats when we pick from the freshmen and sophomores only.
To recap everything so far:
4 ways to pick the 1 junior and 1 senior10 ways to pick the 3 other students (freshmen + sophomores)Therefore, we have 4*10 = 40 different combinations possible for case A. We'll refer to this value later.
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Case B: We pick 2 juniors and 2 seniors
Since there 2 juniors to pick from, and 2 junior seats to fill, there's only 1 way to do this. Likewise, there's only 1 way to pick the 2 seniors to fill the 2 seats.
In total so far there is 1*1 = 1 way to pick the 2 juniors and 2 seniors in any order you like.
Then we have 9-4 = 5 students left that are freshmen or sophomores. This is the number of choices we have for the final 5th seat.
We have 1*5 = 5 ways to have case B happen.
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Summary so far:
40 ways to do case A5 ways to do case B40+5 = 45 ways to do either case.There are 45 ways to have the same number of juniors as seniors (either 1 of each or 2 of each).
Now we must calculate the number of total combinations possible on this committee. We'll turn to the nCr formula again.
n = 9 students
r = 5 seats
n C r = (n!)/(r!(n-r)!)
9 C 5 = (9!)/(5!*(9-5)!)
9 C 5 = (9!)/(5!*4!)
9 C 5 = (9*8*7*6*5!)/(5!*4!)
9 C 5 = (9*8*7*6)/(4!)
9 C 5 = (9*8*7*6)/(4*3*2*1)
9 C 5 = (3024)/(24)
9 C 5 = 126
There are 126 different five-person committees possible.
Of those 126 committees, 45 of them consist of cases where we have the same number of juniors and seniors.
That must mean there are 126-45 = 81 combinations where we do not have the same number of juniors and seniors. This is where the concept of "complement" comes in.