The coefficient of x ⁡ 1 0 1 2 in the expansion of 1 + x n + x 253 1 0 , (where n ≤22 is any positive integer), is

Step-by-step solution

Given 1 + x n + x 253 1 0 = 1 + x 2 5 3 + x n 1 0 Using the binomial expansion a + b n = C 0 n a n b 0 + C 1 n a n - 1 b + C 2 n a n - 2 b 2 + . . . + C n n a 0 b n , = 1 0 C 0 1 + x 2 5 3 1 0 x n 0 + 1 0 C 1 1 + x 2 5 3 9 x n 1 + 1 0 C 2 1 + x 2 5 3 8 x n 2 + . . . + 1 0 C 1 0 1 + x 2 5 3 0 x n 1 0 As 2 5 3 = 2 3 × 1 1 and 1 0 1 2 = 2 5 3 × 4 , also n ≤22 ⇒ Coefficient of x 1012 will come only from the first term, i.e. in 10 C 0 1 + x 2 5 3 1 0 x n 0 = 1 + x 253 10 The general term in the exp