Question

Answer the questions below.(a) A certain committee consists of 16 people. From thecommittee, a president, a vice-president, asecretary, and a treasurer are to be chosen. In how many ways can these 4 offices be filled? AssumeXthat a committee member can hold0at most one of these offices.(b) A company has 39 salespeople. A boardsalespeople, ranked in order of effectiveness. How many such rankings are possible? member at the company asks for a list of the a top 40

Answer 1

Since the committee is going to choose the president, vice-president, secretary, and treasurer from the 16 people that conform it, and 1 person cannot be in more than 1 charge, it means that order will matter in the calculation and we will need to use combination instead of permutations.

The general formula for a combination is:

$nCk=\frac{n!}{k!(n-k)!}$

then, if we replace the 16 people as n, and 4 as the 4 charges that need to be found as k, we obtain:

$\begin{gathered} 16C4=\frac{16!}{4!(16-4)!} \\ 16C4=\frac{16!}{4!12!} \end{gathered}$

simplify the expression,

$\begin{gathered} 16C4=\frac{16\ast15\ast14\ast13}{4\ast3\ast2\ast1} \\ 16C4=\frac{43680}{24} \\ 16C4=1820 \end{gathered}$

Answer:

The committee can be arranged in 1820 ways.