Problem: 12 students need to form five study groups. They will form three study groups with 2 students each and two study groups with 3 students each. In how many ways can these groups be formed? Solution: 12 students need to form five study groups: three groups of size 2 and two groups of size 3. We can count these by first lining up all 12 students in any order (giving $12!$ possibilities), then taking the first two as one pair, the next two as another, the next two as the third pair, the next three as one triple, and the last three as the final triple; since the order inside each group doesn't matter we divide by $(2!)^3(3!)^2$, and because the three pairs themselves are indistinguishable, we divide by $3!$, and the two triples are indistinguishable so we divide by $2!$, giving altogether the result $\frac{12!}{(2!)^3(3!)^2\,3!\,2!}$.