Question

A committee of three is chosen from a group of 20 people. How many different committees are possible, if

(a) the committee consists of a president, vice president, and treasurer?

(b) there is no distinction among the three members of the committee?

Answer 1

Answer on Question #75438 – Math – Discrete Mathematics

Question

A committee of three is chosen from a group of 20 people. How many different committees are possible, if

(a) the committee consists of a president, vice president, and treasurer?

(b) there is no distinction among the three members of the committee?

Solution

(a) The number of arrangements without repetitions is equal to the number of $k$-combinations multiplied by the number of permutations between them $C_n^k \cdot k! = \frac{n!}{k!(n-k)!} \cdot k! = \frac{n!}{(n-k)!}$.

In our case $n = 20; k = 3; \frac{20!}{17!} = 18 \cdot 19 \cdot 20 = 6840$.

(b) A $k$-combination of a set $S$ is a subset of $k$ distinct elements of $S$. If the set has $n$ elements, then the number of $k$-combinations is equal to the binomial coefficient: $C_n^k = \frac{n!}{k!(n-k)!}$.

In our case $n = 20; k = 3; C_{20}^3 = \frac{20!}{3! \cdot 17!} = \frac{18 \cdot 19 \cdot 20}{6} = 1140$.

Answer: (a) 6840; (b) 1140.

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