Passage

A cricket team of 11 players is to be performed from a group of 15 players- A.B,C,D,E,F,G,H,I,J,K,L,M,N,O. Among the players A,D,K,L,M,N and O are batsman ;B,C,E,F,G and H are bowlers; I and J are wicketkeepers. It is also known that. I. The team must have at least 5 batsmen and exactly 1 wicketkeeper. II. H can be selected only if B is selected. III. F can be selected only if both G and N are selected. IV. If I is selected, then F is also selected. V. K and M cannot be selected together for the team. The same is true for B and G.

In how many ways can the team be formed?

Checking the conditional rules reveals contradictions when a second wicketkeeper is added or specific bowlers are chosen. If wicketkeeper I is selected, rule IV forces F to be selected, which by rule III forces both G and N to be selected. However, rule V prohibits B and G from being together, and rule II requires B if H is selected, making it impossible to reach exactly 11 players without breaking a rule. Similarly, any combination using H or E requires B, which restricts G, making it impossible to add F or I, and forcing the remaining bowlers to be C and D. The remaining valid combinations are formed by selecting 6 batsmen from the 7 available (since K and M cannot be chosen together), paired with C and D, yielding exactly 6 valid teams.

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