CMI2010-A-02

Question

We need to choose a team of $11$ from a pool of $15$ players and also select a captain. The number of different ways this can be done is

• A. $ \begin{pmatrix} 15 \\ 11 \end{pmatrix}$
• B. $11$ . $ \begin{pmatrix} 15 \\ 11 \end{pmatrix}$
• C. $15 . 14 . 13 . 12 . 11 .10 . 9 . 8 . 7 . 6 . 5$
• D. $(15 . 14 . 13 . 12 . 11 .10 . 9 . 8 . 7 . 6 . 5) . 11$

Answer 1

Number of ways selecting a captain from $15$ players $= \begin{pmatrix} 15 \\ 1 \end{pmatrix}$

Number of ways selecting remaining team members from remaining $14$ players $=\begin{pmatrix} 14 \\ 10 \end{pmatrix}$  

The number of different ways to choose a team of $11$ from a pool of $15$ players and

also select a captain,

                                         $= \begin{pmatrix} 15\\ 1 \end{pmatrix}$ *$ \begin{pmatrix} 14 \\ 10 \end{pmatrix}$

                                         $ =15*13*11*7$ 

                                         $= 11 * \begin{pmatrix} 15 \\ 11 \end{pmatrix}$

Hence,Option (B) is the correct choice.

Comments

Why we have not selected a captain from the 11 players selected and why we are first bothering about the captain?

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You can do as your method answer will be same:

select 11 player from 15 = (¹⁵c₁₁)

Now from 11 any one can be captain = 11 possibility 

Total ways =  (¹⁵c₁₁)* 11

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Thanks Anu007

Answer 2

We can divide the task (S) into 2 sub-task.

S1- Choosing a team of 11 from 15 players

S2- Now selecting a caption from that team.

After that required no. of ways can be obtained by applying product rule.

S= S1.S2

Choosing a team of 11 from 15 players can be done in $\binom{15}{11}$ ways.

Now selecting a caption from that team of 11 players can be done in 11  ways.

So total no, of ways=  11.$\binom{15}{11}$ 

So B is the correct option.

Comments

Option B and D are same...then which option to select?

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 pallaviamu

how B and D are same??

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@ pallaviamu D is ($15_P$₁₁)11 but we have to do comination not permutation . So, B.

Answer 3

choose a team of 11 from a pool of 15 players

select 11 players from 15 players =  $\binom{15}{11}$ ways

also select a captain

we have to select a captain from the 11 selected members and it can be done in   $\begin{pmatrix}11 \\ 1\end{pmatrix}$ ways = 11 ways.

The number of different ways this can be done is

i.e. the total combinations ( ways) possibe are   11 * $\begin{pmatrix}15 \\ 11\end{pmatrix}$ ways