Eleven players are to be selected for a match out of an available list

This is a question on combinations since we have to select a certain number of players from a pool. Therefore, answer option B can be eliminated straight away since \(12P9\) does not represent combinations.

The pool has 14 players. This would have been the value of n had there been no other constraints. However, we know that the best two players will always be selected; this means that these two people will be there in the team no matter what happens.

Our team should have a total of 11 players of which we know that the presence of the 2 best players is a given.

Therefore, we now need to only select 9 players out of a total of 12 available players – note that the pool is no longer 14, because the two best players are no longer part of the pool since they are already a part of the team.

Number of teams that can be formed = \(12C9\).

The correct answer option is C.

Out of n objects, if we know that k objects are always selected, then the pool will reduce by k and the number of objects to be selected will also reduce by k.
Effectively, number of selections = \((n-k)C(r-k)\)

Hope that helps!
Aravind B T

Ideally, we have to only select 9 players out of 12 available players as 2 of the players are always selected.

This can be done in \(^{12}{C_9}\) ways.

Answer C

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