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.