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Question 3: Mathematical Induction Part 1 (a) Suppose a school consists of exactly n 2 2 students, where n is an even number. The students would like to pair up into groups of size two in order to participate in a doubles tennis match. Let a, be the number of ways of pairing up these n students. Show by induction that n! 20/2(n/2)! (b) The similar school as above is offering a set of extra curricular activities. There are n 2 5 students in the school, where a is a multiple of 5, and exactly n/5 different extra-curricular activities. The director must assign exactly 5 students to each extra-curricular activity, where each student participates in exactly one activity. Let b,, be the number of ways of doing this. Show by induction that n! ba = 120n/5