Finding a and b? AIME Problem

Probability Level 3

In the expansion of $(ax+b)^{2000}$ , where $a$ and $b$ are relatively prime positive integers, the coefficients of $x^2$ and $x^3$ are equal.

Find $a+b$ .

1 solution

Hana Wehbi
Jun 23, 2017

Using the binomial theorem $\binom{2000}{1998} b^{1998}a^2 = \binom{2000}{1997}b^{1997}a^3 \implies b= 666a$ .

Since $a$ and $b$ are relatively prime $\implies a=1$ and $b=666$ .

$a+b= 1+666= 667.$