Question

Any five people on a committee of nine can decide for the committee. How many groups of five people are there?

15,120

1,860

45

126

Answer 1

To find the number of groups of five people that can be formed from a committee of nine, we can use the combination formula:

\[
\binom{n}{r} = \frac{n!}{r!(n-r)!}
\]

where \( n \) is the total number of people, \( r \) is the number of people to choose, and \( ! \) denotes factorial. Here, \( n = 9 \) and \( r = 5 \).

Thus, we need to calculate:

\[
\binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9!}{5! \cdot 4!}
\]

Calculating the factorials:

- \( 9! = 9 \times 8 \times 7 \times 6 \times 5! \)
- Hence, \( 9! / 5! = 9 \times 8 \times 7 \times 6 \)

Now we can simplify the expression:

\[
\binom{9}{5} = \frac{9 \times 8 \times 7 \times 6}{4!}
\]

Calculating \( 4! = 4 \times 3 \times 2 \times 1 = 24 \):

Now substituting this back, we have:

\[
\binom{9}{5} = \frac{9 \times 8 \times 7 \times 6}{24}
\]

Calculating the numerator:

\[
9 \times 8 = 72 \\
72 \times 7 = 504 \\
504 \times 6 = 3024
\]

Now, dividing by \( 24 \):

\[
\binom{9}{5} = \frac{3024}{24} = 126
\]

Thus, the number of groups of five people that can be formed from a committee of nine is:

\[
\boxed{126}
\]