Compute permutations with repeated letters

Given two nonnegative integers m and n, how many distinct strings can be formed that contain exactly m copies of the character 'a' and n copies of the character 'b'? Provide the closed-form count and a short justification. Then generalize your answer to an alphabet with k distinct characters occurring with counts c1, c2, ..., ck (sum ci = N), and give the corresponding formula.