Please answer both Question 2 & 3 please!

Question 2 (Cycle Structure in Canonical Cycle Notation, 20 points). Given positive integers a, 22, ..., ak with ai + a2 + ... + ak = n, how many permutations of [n] when written in canonical cycle notation have cycles of length ai, a2, ..., ak in that order? Question 3 (Permutations with Large Cycles, 30 points). Let n be a positive integer. (a) Let k > n/2. Give a formula for the number of permutations of [n] that contain a cycle of length exactly k. Hint: you will want to count something in two different ways here. [25 points) (6) If n is large, approximately what fraction of permutation of [n] contain a cycle of length more than n/2? [5 points) Question 2 (Cycle Structure in Canonical Cycle Notation, 20 points). Given positive integers a, 22, ..., ak with ai + a2 + ... + ak = n, how many permutations of [n] when written in canonical cycle notation have cycles of length ai, a2, ..., ak in that order? Question 3 (Permutations with Large Cycles, 30 points). Let n be a positive integer. (a) Let k > n/2. Give a formula for the number of permutations of [n] that contain a cycle of length exactly k. Hint: you will want to count something in two different ways here. [25 points) (6) If n is large, approximately what fraction of permutation of [n] contain a cycle of length more than n/2? [5 points)