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 atmost $1$ all rounder and $1$ wicket keeper will play is:

• A. ${{}^4}_{C_1}.{{}^{14}}_{C_{10}}+{{}^2}_{C_1}.{{}^{14}}_{C_{10}}+{{}^4}_{C_1}.{{}^2}_{C_1}.{{}^{14}}_{C_9}+{{}^{14}}_{C_{11}}$
• B. ${{}^4}_{C_1}.{{}^{15}}_{C_{11}}+{{}^{15}}_{C_{11}}$
• C. ${{}^4}_{C_1}.{{}^{15}}_{C_{10}}+{{}^{15}}_{C_{11}}$
• D. ${{}^4}_{C_1}.{{}^2}_{C_1}.{{}^{14}}_{C_9}+{{}^{14}}_{C_{11}}$

Answer 1

The correct option is A ${{}^4}_{C_1}.{{}^{14}}_{C_{10}}+{{}^2}_{C_1}.{{}^{14}}_{C_{10}}+{{}^4}_{C_1}.{{}^2}_{C_1}.{{}^{14}}_{C_9}+{{}^{14}}_{C_{11}}$

When $1$ all rounder and $10$ players from bowlers and batsman play number of ways $={{}^4}_{C_1}.{{}^{14}}_{C_{10}}$
When $1$ wicket keeper and $10$ players from bowlers and batsman play number of ways
$={{}^2}_{C_1}.{{}^{14}}_{C_{10}}$
When $1$ all rounder $1$ wicket keeper and $9$ from batsman and bowlers play number of ways
$={{}^4}_{C_1}.{{}^2}_{C_1}.{{}^{14}}_{C_9}$
When all eleven players play from bowlers and batsman then the number of ways $={{}^{14}}_{C_{11}}$
$\therefore$ total number of selections $={{}^4}_{C_1}.{{}^{14}}_{C_{10}}+{{}^2}_{C_1}.{{}^{14}}_{C_{10}}+{{}^4}_{C_1}.{{}^2}_{C_1}.{{}^{14}}_{C_9}+{{}^{14}}_{C_{11}}$.