Solution:
We select exactly I wicket keeper from 4 .
So, number of ways =4C1=4
Now, we have 10 players left to select from batsmen, bowlers and all-rounders with the condition.
Batsmen ≥4, Bowlers ≥3, All-rounders ≥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 .
∴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
=(6C46C44C2+6C46C34C3+6C56C34C2)
=(15×15×6+15×20×4+6×20×6)
=(1350+1200+720)=3270
∴ Total number of ways for 11 players
=4×3270
=13080
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