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Ms. Bell's mathematics class consists of 8 sophomores, 13 juniors, and 9 seniors. How many different ways can Ms. Bell create a 3-member committee of sophomores if each sophomore has an equal chance of being selected?
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Ms. Bell's mathematics class consists of $8$ sophomores, $13$ juniors, and $9$ seniors. How many different ways can Ms. Bell create a $3$ -member committee of sophomores if each sophomore has an equal chance of being selected? $\newline$ Answer:

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Probability of simple events

Full solution

Q. Ms. Bell's mathematics class consists of

8

sophomores,

13

juniors, and

9

seniors. How many different ways can Ms. Bell create a

3

-member committee of sophomores if each sophomore has an equal chance of being selected?

\newline

Answer:

Define Combination Formula: To determine the number of different ways to create a $3$ -member committee from the $8$ sophomores, we need to use the combination formula, which is defined as $C(n, k) = \frac{n!}{k!(n - k)!}$ , where $n$ is the total number of items to choose from, $k$ is the number of items to choose, and “ $!$ ” denotes factorial. $\newline$ In this case, $n = 8$ (the number of sophomores) and $k = 3$ (the number of members for the committee).
Calculate Factorial of $n$ : First, we calculate the factorial of $n$ , which is $8!$ ( $8$ factorial). $8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320$
Calculate Factorial of $k$ : Next, we calculate the factorial of $k$ , which is $3!$ ( $3$ factorial). $3! = 3 \times 2 \times 1 = 6$
Calculate Factorial of $(n - k)$ : Then, we calculate the factorial of $(n - k)$ , which is $(8 - 3)!$ or $5!$ ( $5$ factorial). $\newline$ $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
Use Combination Formula: Now, we can use the combination formula to find the number of different ways to create the committee: $C(8, 3) = \frac{8!}{3!(8 - 3)!} = \frac{40,320}{6 \times 120}$
Simplify Calculation: We simplify the calculation by dividing $40,320$ by $6$ and then by $120$ : $\newline$ $40,320 / 6 = 6,720$ $\newline$ $6,720 / 120 = 56$
Final Result: Therefore, there are $56$ different ways Ms. Bell can create a $3$ -member committee of sophomores.

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