Enumerative Combinatorics

Practice problems

Try a few Enumerative Combinatorics problems, each with a hint.

• A group has six distinct volunteersCounting principles, bijections, and the pigeonhole principle
• Among n distinct volunteers, a specified set of r volunteers is qualified to serve as…Counting principles, bijections, and the pigeonhole principle
• Let a,b ≥ 0 and let k be an integerBinomial and multinomial coefficients, identities, and lattice paths
• Let a,b ≥ 0 and h ≥ 1 be integers with b-a<hBinomial and multinomial coefficients, identities, and lattice paths
• For n ≥ 1, a positive composition of n is an ordered list of positive integers summing…Compositions, multisets, and distributing objects into boxes
• Among thirty labeled projects, eighteen use method A, fourteen use method B, and twelve…Inclusion-exclusion, derangements, and surjections
• Let S(n,k) count partitions of [n] into k nonempty unordered blocksSet partitions, Stirling numbers, and the cycle structure of permutations
• An integer partition of n is a finite weakly decreasing sequence of positive integers…Integer partitions and Ferrers diagrams
• Let a₀=2, a₁=5, and aₙ=5aₙ₋₁-6aₙ₋₂ for n ≥ 2Ordinary generating functions, recurrences, and Catalan numbers
• If aₙ counts structures on a specified set of n distinct labels, its exponential…Exponential generating functions and labeled structures