Question

How many ways are there to pick a group of n people from 100 people, and then pick a second group of m people from the remaining 100- n people, so that all the people you chose from the first group are taller than the people from the second group (assume that everyone in the group of 100 people has a different height) ote: it is implied in this question that 1 <n < 100 an d 0m100-n.

Answer 1

The first group are to be formed of n people from 100 people and the second group comprised of m people from the remaining 100-n people. The group is formed such that people of the first group are taller than the people from the second group. Since the n people in the first group is taller than the people form the second group, therefore, the number of ways of selecting the group is 1. The first group is comprised of n members, so the number of arrangements of the first group is n!. Thus, the total number of ways of selecting the people in the first group is n!, 1sns 100 Since the second group of m is formed from remaining 100-n people, therefore, the (100-m number of ways of selecting the second group is , 0<m<100-n and (100- m I m m (100 – m)! !(100 – 2m)! The total number of ways of picking the two groups is the sum of the number of ways of selecting the first group and the number of ways of selecting the second group. (100-m)! Hence, the total number of ways of selecting the two groups is n!+- , where, m!(100-2m)! 1sns100 and 0 <m <100-n.