Triangles Are Important Here

Relevant wiki: Binomial Theorem

Solving this question is equivalent to solving the equation $\dbinom{2n+1}3 + \dbinom{2n+1}4 = \dbinom{2n+2}n$ .

And by definition of binomial coefficient , $\dbinom NK = \dbinom N{N-K}$ , so $\dbinom{2n+2}n = \dbinom{2n+2}{(2n+2) - n} = \dbinom{2n+2}{n+2}$

Using the properties of binomial coefficients , (or more commonly known as Pascal's rule ), we have

$\dbinom NK + \dbinom N{K+1} = \dbinom{N+1}{K+1} .$

In this case, $N = 2n+1, K = 3$ , so $\dbinom{2n+1}3 + \dbinom{2n+1}4 = \dbinom{2n+2}4$ . Thus
$\dbinom{2n+2}n = \dbinom{2n+2}4 \Rightarrow n = 4$ , or
$\dbinom{2n+2}{n+2} = \dbinom{2n+2}4 \Rightarrow n+2 = 4 \Rightarrow n =2$ .

Hence, the values of $a$ and $b$ are 2 and 4. And their sum is $2+4 =\boxed6$ .