Writing $\binom{m}{n} - \binom{m-a}{n}$ as one binomial coefficient, $m > n$ and $m-a > n$

Question

I am dealing with these type of expressions for combinatorics. I am wondering if there is another way of writing this as a unique combinatorial number of type $\binom{p}{q}$:

$$\binom{m}{n} - \binom{m-a}{n}$$

Answer 1

$$\binom{\binom{m}{n} - \binom{m-a}{n}}{1}$$

Answer 2

I don't know if it will help, but it is the sum of $a$ terms $${m-1\choose n-1}+{m-2\choose n-1}+...+{m-a\choose n-1}$$
If you want to prove it, it is quite easy by induction.