The number of four-digit numbers, all four of whose digits are distinct and have sum 10, is

Find the sets of four distinct digits with sum $10$. If $0$ is absent the digits are $\{1,2,3,4\}$ only, since the smallest possible sum is $10$. If $0$ is present, the other three distinct non-zero digits sum to $10$: $\{1,2,7\},\{1,3,6\},\{1,4,5\},\{2,3,5\}$. For $\{1,2,3,4\}$ all $4!=24$ arrangements are valid four-digit numbers. For each of the four sets containing $0$, the leading digit cannot be $0$, giving $3\cdot3!=18$ numbers. The total is $24+4\cdot18=96$.