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The principal would like to assemble a committee of 6 students from the 14-member student council. How many different committees can be chosen?
Answer:

The principal would like to assemble a committee of $6$ students from the $14$ -member student council. How many different committees can be chosen? $\newline$ Answer:

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Compound events: find the number of outcomes

Full solution

Q. The principal would like to assemble a committee of

6

students from the

14

-member student council. How many different committees can be chosen?

\newline

Answer:

Identify Problem Type: Identify the type of problem and the formula to use. $\newline$ We need to find the number of ways to choose $6$ students out of $14$ without regard to order. This is a combination problem, and the formula for combinations is: $\newline$ $C(n, k) = \frac{n!}{k!(n - k)!}$ $\newline$ where $n$ is the total number of items, $k$ is the number of items to choose, and $!$ denotes factorial.
Apply Formula: Apply the formula to the given numbers. $\newline$ We have $n = 14$ (total members) and $k = 6$ (members to choose for the committee). Plugging these values into the formula gives us: $\newline$ $C(14, 6) = \frac{14!}{(6!(14 - 6)!)}$ $\newline$ $= \frac{14!}{(6!8!)}$
Calculate Factorials: Calculate the factorials and simplify the expression. $\newline$ $14!$ means $14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8!$ $\newline$ $6!$ means $6 \times 5 \times 4 \times 3 \times 2 \times 1$ $\newline$ $8!$ cancels out in the numerator and denominator. $\newline$ So, we have: $\newline$ $C(14, 6) = \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9}{6 \times 5 \times 4 \times 3 \times 2 \times 1}$
Perform Calculations: Perform the calculations. $\newline$ We can simplify the fraction by canceling common factors before multiplying to avoid large numbers: $\newline$ $C(14, 6) = \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9}{6 \times 5 \times 4 \times 3 \times 2 \times 1}$ $\newline$ $C(14, 6) = \frac{7 \times 13 \times 3 \times 11 \times 10 \times 3}{5 \times 2 \times 1}$ $\newline$ $C(14, 6) = 7 \times 13 \times 11 \times 10 \times 3 \times 3$ $\newline$ $C(14, 6) = 3003$

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