How many different license plates start with the letter A if letters and digits cannot be repeated? Enter the exact numeric answer.

Question

a) Consider the equation $x_1 + x_2 + x_3 + x_4 = 35$. How many different solutions does this equation have if all the variables must be positive integers? Enter the exact numeric answer.

b) Suppose that a license plate consists of three letters followed by three digits. How many different license plates start with the letter A if letters and digits cannot be repeated? Enter the exact numeric answer.

My work

a) $C(38,35)$

b)$1*P(26,2)*P(10,3)$

A_ _ _ _ _ . The last three number should not be repeated, so $P(10,3)$ and then the letters can be chosen randomly so $C(26,2)$

Answer 1

(a) Almost. The restriction is that all variables must be positive integers, so we can rewrite this as

$$ y_1+y_2+y_3+y_4 = 31 $$

with $y_i = x_i-1$ being a non-negative integer for all $i$. Then stars-and-bars gives us $C(34, 31) = C(34, 3)$.

(b) Almost. As Tony Jacobs points out in the comments, the two letters should be represented by $P(25, 2)$, since you have to avoid $A$ (leaving only $25$ letters left to chose from), and order matters, as it does with the digits.

Answer 2

Hint for a)

Write $$35=1+1+\dots+1$$ Now, in how many ways can you place three commas? For instance $1,1+1,1+1+1,1+\dots+1$.

For b), you are not considering that the digits come after the letters.