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Eleven players are to be selected for a match out of an available list 10 Jun 2021, 06:45
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C
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76% (00:58) correct 24% (01:12) wrong based on 213 sessions

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Eleven players are to be selected for a match out of an available list of 14 players. In how many ways can this be done such that the best two identified players are always selected?

(A) 14C11
(B) 12P9
(C) 12C9
(D) 12C9 × 2!
(E) 12C11
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C
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Eleven players are to be selected for a match out of an available list 11 Jun 2021, 13:06
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Bunuel
Eleven players are to be selected for a match out of an available list of 14 players. In how many ways can this be done such that the best two identified players are always selected?

(A) 14C11
(B) 12P9
(C) 12C9
(D) 12C9 × 2!
(E) 12C11
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Since, 2 players are always selected. We can simply remove them from our calculation.
So, now available players = 12
remaining seats = 9
\(12C_9\) = 220, (C) IMO!
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Eleven players are to be selected for a match out of an available list 12 Jun 2021, 01:44
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Bunuel
Eleven players are to be selected for a match out of an available list of 14 players. In how many ways can this be done such that the best two identified players are always selected?

(A) 14C11
(B) 12P9
(C) 12C9
(D) 12C9 × 2!
(E) 12C11
Show more

Solution :

We are given,
    • Total number of players = 14
    • Number of players to be selected for the match = 11
We need to find the number of ways the selection has to be done such that the best two players are always selected.

    • We first keep the best two players fixed in the team.
    • Then, the remaining players out of the total = 14 - 2 = 12
    • Now, out of these remaining 12 players, we need to select 9 players (since 2 best players are already selected and kept fixed)

Therefore, the required number of ways = 12C9

Hence, the correct answer is Option C.
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Eleven players are to be selected for a match out of an available list 12 Jun 2021, 10:39
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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!
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Eleven players are to be selected for a match out of an available list 12 Jun 2021, 22:00
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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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Eleven players are to be selected for a match out of an available list 10 Sep 2025, 12:15
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