Question

The number of ways a cricket team of 11 players be chosen out of 15 players if whom 6 are bowlers if the team consists of at least of 5 bowlers
(a) 630
(b) 504
(c) 126
(d) 526

Answer 1

We solve this problem by dividing the question in two parts. First part consists of 5 bowlers and 6 batsmen and the second part consists of 6 bowlers and 5 batsmen because at least 5 bowlers mean 5 or more bowlers. Then we use the permutations and combinations to find the required result. The number of ways of selecting $'r'$ members from $'n'$ members is given as ${}^{n}{{C}_{r}}$ where,
${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$

Complete step-by-step answer:
We are given that there are a total of 15 players of which 6 are bowlers.
So, we can say that there are 9 batsmen in the 15 players.
We are asked to find the number of ways of selecting 11 players consisting of at least 5 bowlers.
Let us divide this into two parts.
(i) Taking 11 players having 5 bowlers.
We know that the number of ways of selecting $'r'$ members from $'n'$ members is given as ${}^{n}{{C}_{r}}$ where,
${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$

By using the above formula, we get the number of ways of selecting 5 bowlers from 6 bowlers as

& \Rightarrow {}^{6}{{C}_{5}}=\dfrac{6!}{5!\left( 6-5 \right)!} \\\ & \Rightarrow {}^{6}{{C}_{5}}=6 \\\ \end{aligned}$$ Here, we can see that we have already selected 5 players for a team of 11 players. Now, let us find the number of ways of selecting 6 batsmen from 9 batsmen By using the above formula we get the number of ways of selecting 6 batsmen from 9 batsmen as $$\begin{aligned} & \Rightarrow {}^{9}{{C}_{6}}=\dfrac{9!}{6!\left( 9-6 \right)!} \\\ & \Rightarrow {}^{9}{{C}_{6}}=\dfrac{9\times 8\times 7}{3\times 2}=84 \\\ \end{aligned}$$ Let us assume that the total number of ways of selecting 5 bowlers and 6 batsmen as $$'x'$$ Here we can see that while combining both the numbers that is the number of ways of selecting 5 bowlers and 6 batsmen is a type of permutation. By using the permutations we get the total number of ways of selecting 5 bowlers and 6 batsmen as $$\begin{aligned} & \Rightarrow x=6\times 84 \\\ & \Rightarrow x=504 \\\ \end{aligned}$$ (ii) Taking 11 players having 6 bowlers. We know that the number of ways of selecting $$'r'$$ members from $$'n'$$ members is given as $${}^{n}{{C}_{r}}$$ where, $${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$$ By using the above formula, we get the number of ways of selecting 6 bowlers from 6 bowlers as $$\begin{aligned} & \Rightarrow {}^{6}{{C}_{6}}=\dfrac{6!}{6!\left( 6-6 \right)!} \\\ & \Rightarrow {}^{6}{{C}_{6}}=1 \\\ \end{aligned}$$ Here, we can see that we have already selected 6 players for a team of 11 players. Now, let us find the number of ways of selecting 5 batsmen from 9 batsmen By using the above formula we get the number of ways of selecting 5 batsmen from 9 batsmen as $$\begin{aligned} & \Rightarrow {}^{9}{{C}_{5}}=\dfrac{9!}{5!\left( 9-5 \right)!} \\\ & \Rightarrow {}^{9}{{C}_{5}}=\dfrac{9\times 8\times 7\times 6}{4\times 3\times 2}=126 \\\ \end{aligned}$$ Let us assume that the total number of ways of selecting 6 bowlers and 5 batsmen as $$'y'$$ Here we can see that while combining both the numbers that is the number of ways of selecting 6 bowlers and 5 batsmen is a type of permutation. By using the permutations we get the total number of ways of selecting 6 bowlers and 5 batsmen as $$\begin{aligned} & \Rightarrow x=1\times 126 \\\ & \Rightarrow x=126 \\\ \end{aligned}$$ Let us assume that the total number of ways of selecting at least 5 bowlers as $$'N'$$ which is the combination of selecting 5 bowlers and 6 bowlers By using the above condition we get $$\begin{aligned} & \Rightarrow N=x+y \\\ & \Rightarrow N=504+126 \\\ & \Rightarrow N=630 \\\ \end{aligned}$$ Therefore, there are 630 ways of selecting at least 5 bowlers to form a team of 11 players from 15 players. **So, the correct answer is “Option (a)”.** **Note:** Students may make mistakes at permutations and combinations. The number of ways of selecting 5 bowlers and 6 batsmen to form a team of 11 players is a permutation type. So, we get $$\Rightarrow x=6\times 84$$ Whereas, selecting at least 5 bowlers is a combination of selecting 5 bowlers and selecting 6 bowlers. So, we get $$\Rightarrow N=x+y$$ Students may make mistakes in understanding the permutations concept and combinations concept.