How many ways can four people be selected at random from a committee of twenty people?

To determine how many ways four people can be selected from a committee of twenty, we use the concept of combinations, as the order in which the individuals are selected does not matter. The formula for combinations is given by:

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

where ( n ) is the total number of items to choose from, ( r ) is the number of items to choose, and "!" denotes factorial, which is the product of all positive integers up to that number.

In this case, we are selecting 4 people from a total of 20. Thus, we set ( n = 20 ) and ( r = 4 ). Now, substituting these values into the combinations formula gives:

[ C(20, 4) = \frac{20!}{4!(20 - 4)!} = \frac{20!}{4! \cdot 16!} ]

This simplifies to:

[ C(20, 4) = \frac{20 \times 19 \times 18 \times 17}{4 \times 3 \times 2 \times 1} ]

Calculating the