= Solution
Pad every <integer> from $0$ through $999999$ with leading zeros to obtain a unique six-digit decimal string. If its digits sum to $5$, each digit is automatically at most $5$, so the usual digit <upper bound> $9$ adds no restriction. We therefore count the nonnegative <integer> solutions of
$$
d_1+d_2+\cdots+d_6=5.
$$
By <stars and bars>, there are $\binom{5+6-1}{6-1}=\binom{10}{5}$. The all-zero string is not counted because its <digit sum> is zero, so every counted <integer> is positive. Thus the answer is \b[$252$].
Back to article page
