Let \(a\) be a fixed object. How many ways are there to distribute 6 distinct objects, including \(a\), into three labeled boxes of 2 objects each, with \(a\) in box 1?
There are 5 choices for the partner of \(a\) and \(\binom{4}{2}=6\) ways to split the other four; \(5\cdot 6=30\). Without the restriction there would be 90.
Multinomial coefficient \(\binom{n}{k_1,\dots,k_m}\): the number of ways to split \(n\) distinct objects into \(m\) labeled groups of sizes \(k_1,\dots,k_m\), where \(k_1+\cdots+k_m=n\).