there are 15 students including john and mary, in the art club of a school a committee of 3 men consists of chairman secretary and treasurer is selected from the club. if john and mary cannot be both selected how many different committees can be formed ?
thank you:)
If the committee is really 3 men, then Mary can't be on it at all. I suspect you meant 3 people. In that case, if either John or Mary is on the committee, then there are only 13 other people to pick for the other two seats. That gives us
Committees with John: 13P2 = 156
Committees with Mary: 13P2 = 156
Committees with neither: 13P3 = 1716
Steve, I was thinking along these lines, can you find a flaw in this?
number of 3-person committees that can be formed from the 15 without any restrictions
= C(15,3) = 455
number of 3-person committees WITH BOTH John and Mary = C(13,1) = 13
so the number without both John and Mary
= 4550- 13
= 442
For each of these , the 3 jobs can be arranged in
3! or 6 ways
number of committees as described that can be formed
= 6(442) = 2652
To find the number of different committees that can be formed, we need to consider two cases:
Case 1: John is in the committee
In this case, we have 13 remaining students (excluding John and Mary) to choose from for the other two positions. Therefore, the number of committees with John in it will be the number of ways to choose 2 students from the remaining 13: C(13, 2) = 13! / (2! * (13-2)!) = 78.
Case 2: Mary is in the committee
In this case, we have the same 13 remaining students to choose from for the other two positions. Therefore, the number of committees with Mary in it will also be 78.
Since these two cases are exclusive (John and Mary cannot both be in the committee), the total number of different committees that can be formed is the sum of the number of committees in each case: 78 + 78 = 156.
Therefore, there are 156 different committees that can be formed.
To solve this problem, we need to consider two scenarios:
Scenario 1: John is selected.
In this case, we need to select 2 more men from the remaining 13 students in the art club. The number of ways to choose 2 men from 13 is given by the combination formula (nCr):
13C2 = 13! / (2! * (13-2)!) = 13! / (2! * 11!) = (13 * 12) / (2 * 1) = 78
Scenario 2: Mary is selected.
Similar to scenario 1, we need to select 2 more men from the remaining 13 students in the art club. The number of ways to choose 2 men from 13 is also 78.
However, we need to subtract the case where both John and Mary are selected together, as it is not allowed. In this case, we need to select 1 more man from the remaining 13 students in the art club. The number of ways to choose 1 man from 13 is given by 13C1 = 13.
Therefore, the total number of different committees that can be formed is:
Total = Scenario 1 + Scenario 2 - Case where both John and Mary are selected
= 78 + 78 - 13
= 143
Hence, there are 143 different committees that can be formed.