Question

In how many ways can a committee of 3 people be chosen from a group of 7 people?(1 point)

Answer 1

To determine how many ways a committee of 3 people can be chosen from a group of 7 people, we use the combination formula, which is given by:

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

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

In this case, \( n = 7 \) and \( r = 3 \). Plugging in these values:

\[
C(7, 3) = \frac{7!}{3!(7-3)!} = \frac{7!}{3! \cdot 4!}
\]

Calculating the factorials:

\[
7! = 7 \times 6 \times 5 \times 4!
\]

Now we can substitute this back into the formula:

\[
C(7, 3) = \frac{7 \times 6 \times 5 \times 4!}{3! \cdot 4!}
\]

The \( 4! \) in the numerator and denominator cancels out:

\[
C(7, 3) = \frac{7 \times 6 \times 5}{3!}
\]

Now, calculating \( 3! \):

\[
3! = 3 \times 2 \times 1 = 6
\]

Thus, we have:

\[
C(7, 3) = \frac{7 \times 6 \times 5}{6}
\]

The \( 6 \) in the numerator and denominator cancels out:

\[
C(7, 3) = 7 \times 5 = 35
\]

Therefore, the number of ways to choose a committee of 3 people from a group of 7 is:

\[
\boxed{35}
\]