Ten identical balls are distributed among four distinct boxes so that every box receives at least 1 and at most 4 balls. Find the number of…

We need the number of integer solutions of $a+b+c+d=10$ with $1\le a,b,c,d\le4$. Put $a'=a-1$, etc.: $a'+b'+c'+d'=6$ with $0\le a'\le3$. Without the upper bound there are $\binom93=84$ solutions. Subtract those where some variable is at least $4$: fixing one variable $\ge4$ leaves $\binom{5}{3}=10$ solutions for each of the $4$ choices, giving $40$. Two variables both $\ge4$ would need at least $8>6$, so there is no overlap. By inclusion-exclusion the count is $84-40=44$. A brute-force enumeration over all $4^4$ tuples confirms $44$.