11 persons decide to spend an afternoon in two groups. A group of them decides to go to a theatre and the remaining decide to play tennis. In how many ways can the group for tennis be formed. If there must be at least four persons in each group?

Explanation:

According to question, Case 1

When 4 persons in tennis and rest in theatre.

Number of ways = 11C4×7C7

= 11×10×9×8 \ 4×3×2×1 = 330

Case II When 5 persons in tennis and rest in theatre.

Number of ways = 11C5×6C6

= 11×10×9×8×7 \ 5×4×3×2×1 = 462

Case Ill When 6 persons in tennis and rest in theatre.

Number of ways = 11C6×5C5 = 462

Case IV When 7 persons in tennis and rest in theatre,

Number of ways = 11C7×4C4 = 330

Therefore, Total Number of ways

= 2(330+462)=2(792)

= 1584