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  • A committee needs to be formed with 6 members from a group of 12 individuals. However, two specific individuals, Alex and Emily, cannot be in the committee together. In how many ways can the commit

A committee needs to be formed with 6 members from a group of 12 individuals. However, two specific individuals, Alex and Emily, cannot be in the committee together. In how many ways can the committee be formed?

Option: 1

652


Option: 2

504


Option: 3

752


Option: 4

402


Answers (1)

best_answer

To calculate the number of ways the committee can be formed, we need to consider two scenarios: Alex is in the committee and Emily is not, and Emily is in the committee and Alex is not.

Scenario 1: Alex is in the committee and Emily is not:

In this case, we need to select 5 more members from the remaining 10 individuals (excluding Alex and Emily). The number of ways to do this is given by the combination formula: \mathrm{C(10,5)=10 ! /(5 ! \times(10-5) !)=252 .}

Scenario 2: Emily is in the committee and Alex is not:

Similar to Scenario 1, we need to select 5 more members from the remaining 10 individuals (excluding Alex and Emily). Again, the number of ways to do this is given by the combination formula: \mathrm{C(10,5)=10 ! /(5 ! \times(10-5) !)=252 .}

Since these two scenarios are mutually exclusive (Alex and Emily cannot be in the committee together), we can simply add the results:

\mathrm{\text{Number of ways = Scenario 1 + Scenario 2 = 252 + 252 = 504.}}

Therefore, there are 504 ways to form a committee with 6 members, given that Alex and Emily cannot be in the committee together.

Hence option 2 is correct.

Posted by

Ritika Kankaria

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