Question

A search committee is formed to find a new software engineer.

Answer 1

Answer:

(a) 1,902,231,808,400

(b) 84

(c) 20

Step-by-step explanation:

In mathematics, the procedure to select k items from n distinct items, without replacement, is known as combinations.

The formula to compute the combinations of k items from n is given by the formula:

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

(a)

Compute the number of ways to select 9 applicants from 100 as follows:

${100\choose 9}=\frac{100!}{9!\cdot(100-9)!}$

        $=\frac{100!}{9!\times 91!}\\\\=\frac{100\times 99\times 98\times 97\times 96\times 95\times 94\times 93\times 92\times 91!}{9!\times 91!}\\\\=\frac{100\times 99\times 98\times 97\times 96\times 95\times 94\times 93\times 92}{9!}\\\\=1902231808400$

(b)

Compute the number of ways to select 6 people from 9 as follows:

${9\choose 6}=\frac{9!}{6!\cdot(9-6)!}$

        $=\frac{9!}{6!\times 3!}\\\\=\frac{9\times 8\times 7\times 6!}{6!\times 3!}\\\\=\frac{9\times 8\times 7}{3!}\\\\=84$

(c)

Compute the number of ways to select top 3 candidates from 6 as follows:

${6\choose 3}=\frac{6!}{3!\cdot(6-3)!}$

        $=\frac{6!}{3!\times 3!}\\\\=\frac{6\times 5\times 4\times 3!}{3!\times 3!}\\\\=\frac{6\times 5\times 4}{3!}\\\\=20$