Question

A group consists of four men and five women. Three people are selected to attend a conference

Answer 1

To determine the number of ways to select a group of three people from a group of four men and five women (totaling nine people), we can use the combination formula:

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

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

In this scenario, we are selecting 3 people from a total of 9 (4 men + 5 women).

Calculating \( C(9, 3) \):

\[
C(9, 3) = \frac{9!}{3!(9 - 3)!} = \frac{9!}{3! \cdot 6!}
\]

Calculating \( 9! \) and \( 6! \) we can simplify:

\[
9! = 9 \times 8 \times 7 \times 6!
\]

Thus, the \( 6! \) cancels out:

\[
C(9, 3) = \frac{9 \times 8 \times 7}{3!}
\]

Now calculating \( 3! \):

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

Continuing with the calculation:

\[
C(9, 3) = \frac{9 \times 8 \times 7}{6} = \frac{504}{6} = 84
\]

Thus, there are **84 ways** to select a group of three people from a group of four men and five women.