Permutations and Combinations

We select exactly I wicket keeper from 4 .So, number of ways $= {}^4 C_1 = 4$Now, we have 10 players left to select from batsmen, bowlers and all-rounders with the condition.Batsmen $\ge 4$, Bowlers $\ge 3$, All-rounders $\ge 2$So, total of these three $=10$So, number of batsmen, $B$ from 4 to 6 , number of bowlers, $L$ (say)from 3 to 6 and number of all-rounder, $A$ (say) from 2 to 4 .$\therefore B+L+A=10$ (after 1 wicket keeper)Now, the possible combinations for, (B, L, A) are$(4,4,2),(4,3,3),(5,3,2)$Total number of ways for 10 players$\;= ( {}^6 C_4 {}^6 C_4 {}^4 C_2 + {}^6 C_4 {}^6 C_3 {}^4 C_3 + {}^6 C_5 {}^6 C_3 {}^4 C_2 )$$\;= (15 \times 15 \times 6 + 15 \times 20 \times 4 + 6 \times 20 \times 6)$$\;= (1350+1200+720) = 3270$∴ Total number of ways for 11 players$\;= 4 \times 3270$$\;= 13080$