A club has three different committees each of which needs a chairperson. How many ways can the chair people be selected if only 10 people are eligible for these positions, and no one can chair more than two of the committees
To determine the number of ways the chairpeople can be selected, we can break down the problem into cases.
Case 1: One person chairs all three committees.
In this case, there are 10 options for the chosen person.
Case 2: Two people chair two committees each, and one person chairs the remaining committee.
In this case, we need to choose 2 people out of 10 who will chair two committees each. This can be done in C(10,2) ways. Once these two people are chosen, there are (10-2) = 8 options for the remaining person.
Case 3: One person chairs two committees, and two people chair one committee each.
In this case, we need to choose 1 person out of 10 who will chair two committees. This can be done in C(10,1) ways. Once this person is chosen, there are (10-1) = 9 options for the remaining two people who will chair one committee each. This can be done in C(9,2) ways.
Now we can calculate the total number of ways:
Total ways = (Case 1) + (Case 2) + (Case 3)
Total ways = 10 + C(10,2) * 8 + C(10,1) * C(9,2)
Let's calculate the answer:
Total ways = 10 + C(10,2) * 8 + C(10,1) * C(9,2)
Total ways = 10 + (10! / (2! * (10-2)!) * 8) + (10! / (1! * (10-1)!) * (9! / (2! * (9-2)!)))
Total ways = 10 + (10! / (2! * 8!) * 8) + (10! / (1! * 9!) * (9! / (2! * 7)!))
Total ways = 10 + (10 * 9 / (2 * 1) * 8) + (10 * 9 * 8 / (1 * 2) * (9 * 8 / (2 * 1)))
Total ways = 10 + (90 / 2 * 8) + (720 / 2 * 36)
Total ways = 10 + 360 + 10,080
Total ways = 10,450
Therefore, there are 10,450 ways to select the chairpeople for the three committees from 10 eligible people.
10 choose 3
=10C3
=10*9*8/(1*2*3)