A committee of 4 persons is to be chosen from 8 persons including Mr.Smith and his wife. Mr.Smith will not join without his wife but Mrs.Smith would join without Mr. Smith. Calculate the number of possible ways the commitee can be formed.
if the positions are not unique, then
assuming both join, 6C2 (since only 2 positions have to be filled from the 6 remaining candidates)
assuming only she joins, 6C3 (3 positions have to be filled from the remaining 6 candidates)
now add the two cases
To solve this problem, we need to consider two scenarios:
Scenario 1: Mr. Smith and his wife are both in the committee.
In this case, we need to choose 2 more members from the remaining 6 people (excluding Mr. Smith and his wife). The number of ways to choose these 2 members can be calculated using combination formula: C(6, 2) = 6! / (2! * 4!) = 15.
Scenario 2: Only Mrs. Smith is in the committee.
In this case, we need to choose 3 more members from the remaining 7 people (excluding Mrs. Smith). The number of ways to choose these 3 members can be calculated using the combination formula: C(7, 3) = 7! / (3! * 4!) = 35.
Since Scenario 1 and Scenario 2 are mutually exclusive (they cannot happen at the same time), we can add the number of ways from each scenario to get the total number of ways to form the committee:
Total number of ways = Scenario 1 + Scenario 2 = 15 + 35 = 50.
So, there are 50 possible ways the committee can be formed.