Let $m$ be the smallest positive integer such that the coefficient of $x^{2}$ in the expansion of $(

Question

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

Professor PC Thomas Classes · Answer

(5) : The coefficient of $x^{2}$ in the expansion of $(1+x)^{2}+(1+x)^{3}+\ldots+$ $(1+m x)^{50}$ is\ ${ }^{2} C_{2}+{ }^{3} C_{2}+\ldots+{ }^{49} C_{2}+{ }^{50} C_{2} m^{2}$\ $={ }^{50} C_{3}+{ }^{50} C_{2} m^{2}$\ It is given to be $(3 n+1){ }^{51} C_{3}$\ Now, $\frac{50 \cdot 49 \cdot 48}{6}+\frac{50 \cdot 49}{2} m^{2}=(3 n+1) \frac{51 \cdot 50 \cdot 49}{6}$\ $\Rightarrow \quad 51 n+1=m^{2}$\ As $51 n+1$ must be perfect square.\ So, we have by inspection, $n=5$.\ $	herefore \quad m=16$\