Question

a committee must be formed with 4 teachers and 7 students. if there are 6 teachers to choose from, and 16 students, how many different ways could the committee be made?

Explanation:

Step1: Calculate ways to choose teachers

We use combinations formula \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 6 \), \( k = 4 \).
\( C(6, 4)=\frac{6!}{4!(6 - 4)!}=\frac{6!}{4!2!}=\frac{6\times5}{2\times1}=15 \)

Step2: Calculate ways to choose students

Using combinations formula with \( n = 16 \), \( k = 7 \).
\( C(16, 7)=\frac{16!}{7!(16 - 7)!}=\frac{16!}{7!9!}=\frac{16\times15\times14\times13\times12\times11\times10}{7\times6\times5\times4\times3\times2\times1}=11440 \)

Step3: Multiply the two results

Total ways = \( C(6, 4)\times C(16, 7)=15\times11440 = 171600 \)

Answer:

171600