In how many ways can a committee of 7 people be chosen from 15 married couples if 2 couples must be on the committee?
To find the number of ways a committee of 7 people can be chosen from 15 married couples, while ensuring that 2 couples are included, we can break down the process into steps:
Step 1: Select 2 couples for the committee
There are 15 couples in total, so we can choose 2 couples from them in C(15, 2) ways. Since we only need the couples and not the individual members yet, we consider them as a single unit for now.
Step 2: Select the remaining 3 individuals from the chosen couples
Once we have selected the 2 couples, we need to choose 3 individuals from each selected couple. For each couple, there are 2 possible individuals to choose as they cannot be selected together due to being part of the same couple.
For each selected couple, we can choose 3 individuals in C(2, 3) = 2 ways.
Hence, the total number of ways to select the remaining 3 individuals is 2 * 2 = 4 ways.
Step 3: Select the remaining 2 individuals from the 13 remaining individuals
After selecting the couples and the remaining 6 individuals, we have 13 individuals remaining. These individuals are from 13 married couples.
To select 2 individuals from these 13, we can choose them in C(13, 2) = 78 ways.
Step 4: Calculate the total number of ways
Finally, we need to multiply the results from each step to obtain the total number of ways:
Total ways = number of ways in Step 1 * number of ways in Step 2 * number of ways in Step 3
= C(15, 2) * (2 * 2) * C(13, 2)
= 105 * 4 * 78
= 32,760
Therefore, there are 32,760 ways to choose a committee of 7 people from 15 married couples with the condition that 2 couples must be on the committee.
There are 15C2 ways to choose the couples. That uses up 4 people, leaving 26 individuals to make up the other 3 places on the committee. So, you have
15C2 * 26C3 = 105 * 2600 = 273,000 ways