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Math  /  Discrete

QuestionCounting - Combinations
A committee of seven-consisting of a chairman, a vice chairman, a secretary, and four other members-is to be chosen from a class of 20 students. In how many ways can this committee be chosen?

Studdy Solution

STEP 1

1. The committee consists of specific roles: a chairman, a vice chairman, a secretary, and four other members.
2. There are 20 students available to choose from.

STEP 2

1. Choose the chairman.
2. Choose the vice chairman.
3. Choose the secretary.
4. Choose the four other members.

STEP 3

Choose the chairman from the 20 students. The number of ways to do this is:
(201)=20 \binom{20}{1} = 20

STEP 4

Choose the vice chairman from the remaining 19 students. The number of ways to do this is:
(191)=19 \binom{19}{1} = 19

STEP 5

Choose the secretary from the remaining 18 students. The number of ways to do this is:
(181)=18 \binom{18}{1} = 18

STEP 6

Choose the four other members from the remaining 17 students. The number of ways to do this is:
(174) \binom{17}{4}
Calculate (174) \binom{17}{4} :
(174)=17×16×15×144×3×2×1=2380 \binom{17}{4} = \frac{17 \times 16 \times 15 \times 14}{4 \times 3 \times 2 \times 1} = 2380
To find the total number of ways to choose the committee, multiply the number of ways to choose each position:
20×19×18×2380 20 \times 19 \times 18 \times 2380
Calculate the product:
20×19×18×2380=162,792,000 20 \times 19 \times 18 \times 2380 = 162,792,000
The number of ways to choose the committee is:
162,792,000 \boxed{162,792,000}

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