Chapter 2.1, Problem 16E

Counting when Order Does Not Matter This is a continuation of Exercise 15. In many situations, the number of possibilities is not affected by order. For example, if a group of 4 people is selected from a group of 20 to go on a trip, then the order of selection does not matter. In general, the number C of ways to select a group of K things from a group of n things is given by

$C=\frac{n!}{k!\left(n-\text{k}\right)}$

if k is not greater than n.

1. How many different groups of 4 people could be selected from a group of 20 to go on a trip?
2. How many groups of 16 could be selected from a group of 20?
3. Your answers in parts a and b should have been the same. Explain why this is true.
4. What group size chosen from among 20 people will result in the largest number of possibilities? How many possibilities are there for this group size?