If 4 dice are rolled once, the number of ways of getting the sum as 10 is K , then the value of K 10 is equal to

Step-by-step solution

x 1 + x 2 + x 3 + x 4 = 10 ; 1 ≤ x i ≤ 6 The desired number is the coefficient of x 10 in x + x 2 + … + x 6 4 = x 4 1 + x + … + x 5 4 or the coefficient of x 6 in 1 + x + x 2 + … + x 5 4 = 1 - x 6 1 - x 4 = 1 - x 6 4 1 - x - 4 = 1 - 4 x 6 + … 1 - x - 4 or the coefficient of x 6 in 1 - x - 4 - 4 × coefficient of x 0 in 1 - x - 4 = 6 + 4 - 1 C 4 - 1 - 4 × 0 + 4 - 1 C 4 - 1 = 9 C 3 - 4 × 1 = 9 × 8 × 7 3 × 2 - 4 = 84 - 4 = 80 ⇒ K = 80 ⇒ K 10