The number of solutions to x + y + z = 10 , where 1 ≤ x , y , z ≤ 6 and x , y , z ∈ N , is equal to

Step-by-step solution

x + y + z = 10 Number of solutions is equal to the coefficient of x 10 in the expansion of x + x 2 + x 3 + . … … + x 6 3 = x 3 1 - x 6 1 - x 3 or the coefficient of x 7 in the expansion of 1 - x 6 3 1 - x - 3 = 1 - 3 x 6 1 - x - 3 = C 7 + 3 - 1 7 - 3 C 1 + 3 - 1 1 = C 9 7 - 3 C 3 1 = 36 - 9 = 27