1 Counting (a) Let n,r be positive integers such that n2 r. Give a combinatorial proof of Tr (b) Use (a) to compute ? i(i-1) -1 (c) Let D(n) be the number of permutations ? of such that for all ?(i)?2 Give a combinatorial proof of D(n+1)-n[D (n-1) +D(n)] for all n 2 2 Hint: Think about first choosing where n+1 maps to. (d) It turns out D(n) -n! (-1* Prove this using Inclusion Exclusion 1 Counting (a) Let n,r be positive integers such that n2 r. Give a combinatorial proof of Tr (b) Use (a) to compute ? i(i-1) -1 (c) Let D(n) be the number of permutations ? of such that for all ?(i)?2 Give a combinatorial proof of D(n+1)-n[D (n-1) +D(n)] for all n 2 2 Hint: Think about first choosing where n+1 maps to. (d) It turns out D(n) -n! (-1* Prove this using Inclusion Exclusion