The number of ways to distribute 30 identical candies among four children C 1 , C 2 , C 3 and C 4 so that C 2 receives atleast 4 and atmost 7 candies, C 3 receives atleast 2 and atmost 6 candies, is equal to

Step-by-step solution

Given, C 1 + C 2 + C 3 + C 4 = 30 Now it has given that 4 ≤ C 2 ≤ 7   &   2 ≤ C 3 ≤ 6 and C 1   &   C 4 can take any value. Finding coefficient of x 30 in x 0 + x 1 + . . . . . . x 30 x 4 + x 5 + . . . . . . x 7 x 2 + . . . x 6 x 0 + . . . . x 30 = 1 - x 31 1 - x 2 x 4 1 - x 4 1 - x x 2 1 - x 5 1 - x Ignoring higher power more than 30 = 1 1 - x 2 x 4 1 - x 4 1 - x x 2 1 - x 5 1 - x = x 6 1 - x 4 1 - x 5 1 - x - 4 = x 6 1 - x 5 - x 4 + x 9 1 - x - 4 = x 6 - x 11 -