Let m be the smallest positive integer such that the coefficient of $x^{2}$ in the expansion of ${(1 + x)}^{2} + {(1 + x)}^{3} + \dots + {(1 + x)}^{49} + {(1 + m x)}^{50}$ is $(3 n + 1)^{51} C_{3}$ for some positive integer n.
Then the value of n is

Question

Let m be the smallest positive integer such that the coefficient of

x^{2}

in the expansion of

{(1 + x)}^{2} + {(1 + x)}^{3} + \dots + {(1 + x)}^{49} + {(1 + m x)}^{50}

is

(3 n + 1)^{51} C_{3}

for some positive integer n.
Then the value of n is

Aicharya · Accepted Answer

Coefficient of x 2 in the expansion of 1 + x 2 + 1 + x 3 + … 1 + x 49 + 1 + m x 50 is 2 C 2 + 3 C 2 + … 49 C 2 + 50 C 2 m 2 = 3 n + 1 51 C 3 ⇒ 3 C 3 + 3 C 2 + … 49 C 2 + 50 C 2 m 2 = 3 n + 1 51 C 3 (Use n C r + n C r + 1 = n + 1 C r + 1 ) ⇒ 50 C 3 + 50 C 2 m 2 = 3 n + 1 51 C 3 50.49.48 6 + 50.49 2 m 2 = 3 n + 1 51.50.49 6 m 2 = 51 n + 1 must be a perfect square By trial ⇒ n = 5 and m = 16 ( M , n ∈ N ) ⇒ n = 5

Let m be the smallest positive integer such that the coefficient of x 2 in the expansion of 1 + x 2 + 1 + x…

Solution