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Using $P(n,m)$, the number of integer partitions of $n$ into exactly $m$parts, which was the subject of an earlier paper, $P(n,m,p)$, the number ofinteger partitions of $n$ into exactly $m$ parts with each part at most $p$,can be computed in $O(n^2)$, and the q-binomial coefficient can be computed in$O(n^3)$. Using the definition of the q-binomial coefficient, some propertiesof the q-binomial coefficient and $P(n,m,p)$ are derived. The q-multinomialcoefficient can be computed as a product of q-binomial coefficients. A formulafor $Q(n,m,p)$, the number of integer partitions of $n$ into exactly $m$distinct parts with each part at most $p$, is given. Some formulas for thenumber of integer partitions with each part between a minimum and a maximum arederived. A computer algebra program is listed implementing these algorithmsusing the computer algebra program of the earlier paper.

Computation of q-Binomial Coefficients with the $P(n,m)$ Integer Partition Function