If a cricket team of $11$ players is to be selected from $8$ batsman, $6$ bowlers, $4$ all-rounders and $2$ wicket keepers, then Number of selections when $2$ particular batsmen don’t want to play when a particular bowler will play is:
A. ${}^{17}{{C}_{10}}+{}^{19}{{C}_{11}}$
B. ${}^{17}{{C}_{10}}+{}^{19}{{C}_{11}}+{}^{17}{{C}_{11}}$
C. ${}^{17}{{C}_{10}}+{}^{20}{{C}_{11}}$
D. ${}^{19}{{C}_{10}}+{}^{19}{{C}_{11}}$

Question

If a cricket team of $11$ players is to be selected from $8$ batsman, $6$ bowlers, $4$ all-rounders and $2$ wicket keepers, then Number of selections when $2$ particular batsmen don’t want to play when a particular bowler will play is:
A. ${}^{17}{{C}_{10}}+{}^{19}{{C}_{11}}$
B. ${}^{17}{{C}_{10}}+{}^{19}{{C}_{11}}+{}^{17}{{C}_{11}}$
C. ${}^{17}{{C}_{10}}+{}^{20}{{C}_{11}}$
D. ${}^{19}{{C}_{10}}+{}^{19}{{C}_{11}}$

Vedantu Content Team · Accepted Answer

Hint: To solve this question first we have to know about the combination. Combination is a selection of items or selection of a set of objects from a collection. In combination, order of selection does not matter. Each possible selection is included in the combination. If any condition is given in the question then while selecting the object we must take care that our selection fulfills the given condition.
As we know that the formula for combination is \[{}^{n}{{C}_{r}}=\frac{n!}{r!\left( n-r \right)!}\]
Where, $n=$ number of items/objects
And $r=$ number of items/objects being chosen at a time.

Complete step by step answer:
Here in the given question total number of items = number of batsman + number of bowlers + number of all-rounders + number of wicket keepers
$n=8+6+4+2=20$
We have given that $11$ Players is to be selected from $8$ batsman, $6$ bowlers, $4$ all-rounders and $2$ wicket keepers
When a particular bowler will play, $2$ particular batsmen don’t want to play the rest of $10$players are selected from $17$ other players.
So, the number of selections will be $={}^{17}{{C}_{10}}$
Also, we have given that If the particular bowler doesn’t play then all $11$ players are selected from the rest of $19$ players.
So, the number of selections will be ${}^{19}{{C}_{11}}$
Total number of selection will be ${}^{17}{{C}_{10}}+{}^{19}{{C}_{11}}$
Option A is the correct answer.

Note: Before solving the questions of combination, carefully read the whole question. Analyze the given conditions carefully and then start collecting data. Here when students analyze the options, they may be confused between option A and option C. So, it is necessary to read the condition given in the question carefully that when the particular bowler doesn't play, the total number of players remain 19. And the student must know the formula of combination to solve the question.