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  • A board of directors needs to be formed with 7 members from a pool of 11 candidates. However, two specific candidates, James and Rachel, cannot be on the board together.

A board of directors needs to be formed with 7 members from a pool of 11 candidates. However, two specific candidates, James and Rachel, cannot be on the board together. In how many ways can the board be formed?

Option: 1

293


Option: 2

856


Option: 3

168


Option: 4

963


Answers (1)

To calculate the number of ways the board can be formed, we need to consider two scenarios: James is on the board and Rachel is not, and Rachel is on the board and James is not.

Scenario 1: James is on the board and Rachel is not:

In this case, we need to select 6 more members from the remaining 9 candidates (excluding James and Rachel). The number of ways to do this is given by the combination formula: \mathrm{C(9,6)=9 ! /(6 ! \times(9-6) !)=84}

Scenario 2: Rachel is on the board and James is not:

Similar to Scenario 1, we need to select 6 more members from the remaining 9 candidates (excluding Rachel and James). Again, the number of ways to do this is given by the combination formula:\mathrm{C(9,6)=9 ! /(6 ! \times(9-6) !)=84}

Since these two scenarios are mutually exclusive (James and Rachel cannot be on the board together), we can simply add the results:

\mathrm{\text{Number of ways = Scenario 1 + Scenario 2 = 84 + 84 = 168.}}

Therefore, there are 168 ways to form a board of directors with 7 members, given that James and Rachel cannot be on the board together.

Hence option 3 is correct.

 

Posted by

Sumit Saini

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