Question

If a cricket team of 11 players is to be selected from 8 batsman, 6 bowlers, 4 all-rounder and 2 wicket keepers, then the number of selections when at most 1 all-rounder and 1 wicket keeper will play is:
A. ${}^4{C_1} \cdot {}^{14}{C_{10}} + {}^2{C_1} \cdot {}^{14}{C_{10}} + {}^4{C_1} \cdot {}^2{C_1} \cdot {}^{14}{C_9} + {}^{14}{C_{11}}$
B. ${}^4{C_1} \cdot {}^{15}{C_{11}} + {}^{15}{C_{11}}$
C. ${}^4{C_1} \cdot {}^{15}{C_{10}} + {}^{15}{C_{11}}$
D. ${}^4{C_1} \cdot {}^2{C_1} \cdot {}^{14}{C_9} + {}^{14}{C_{11}}$

Answer 1

Combination is the different selections of a given number of elements taken one by one, or some, or all at a time. For example, if we have two elements A and B, then there is only one way to select two items, we select both of them.
Number of combinations when ‘r’ elements are selected out of a total of ‘n’ elements is
${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$, which can also be represented by ${}^n{C_r} = {}^n{C_{n - r}}$.

Complete step-by-step answer:
Given that a cricket team of 11 players is to be selected from 8 batsman, 6 bowlers, 4 all-rounder and 2 wicket keepers,
Number of players = 11,
Number of batsmen = 8,
Number of all-rounder=4,
Number of wicket-keepers=2,
Now we have to find the number of arrangements when at most 1 all-rounder and 1 wicket keeper,
Number of ways when 1 all-rounder from total of 4, and 10 players from both bowlers and batsman from a total of 14 players are arranged
$\Rightarrow {}^4{C_1} \cdot {}^{14}{C_{10}}$,
Number of ways when 1 wicket-keeper from total of 2, and 10 players from both bowlers and batsman from a total of 14 players are arranged
$\Rightarrow {}^2{C_1} \cdot {}^{14}{C_{10}}$,
Now one wicket-keeper, one all-rounder and 9 players from both bowlers and batsman are arranged in
$\Rightarrow {}^4{C_1} \cdot {}^2{C_1} \cdot {}^{14}{C_9}$ways.
Now when all eleven players are selected from bowlers and batsmen then number of ways these can be arranged $= {}^{14}{C_{11}}$,
Now adding all the arrangements we get the required number of selections, i.e.,
${}^4{C_1} \cdot {}^{14}{C_{10}} + {}^2{C_1} \cdot {}^{14}{C_{10}} + {}^4{C_1} \cdot {}^2{C_1} \cdot {}^{14}{C_9} + {}^{14}{C_{11}}$.

$\therefore$The total number of ways the number of 11 players are selections when at most 1 all-rounder and 1 wicket keeper will play, when there are 8 batsman, 6 bowlers, 4 all-rounder and 2 wicket keepers and should be selected from 11 players is${}^4{C_1} \cdot {}^{14}{C_{10}} + {}^2{C_1} \cdot {}^{14}{C_{10}} + {}^4{C_1} \cdot {}^2{C_1} \cdot {}^{14}{C_9} + {}^{14}{C_{11}}$.

Note:
As the question is related to combinations, we should know the definition and the formula related to the combinations and students should understand the question, and the condition given, as they may get confused in finding the arrangements, which should be done according to the condition given in the question.