Consider the following combinatorial identity: H HE: _ nlk=1 (:1) Present a combinatorial argument for this identity by considering a set of :1 people and determining, in two ways,the number of possible selections of a committee of any size and a chairperson for the committee.Hint: (i) How many possible selections are there of a commit-tee of size k and its chairperson? (ii) How many possible selections are there of a chair-person and the other conunittee members? (b) Verify the following identity for n, = 1, 2, 3, 4, S: Z(:)k2=2_2n(n + 1) k=1 For a combinatorial proof of the preceding, consider a setof :1 people and argue that both sides of the identity rep- resent the number of different selections of a committee,its chairperson, and its secretary (possibly the same as the chairperson).Hint: (i) How many different selections result in the commit-tee containing exactly k people? (ii) How many different selections are there in whichthe chairperson and the secretary are the same?(ANSWER: n21 .) (iii) How many different selections result in the chairper-son and the secretary being different? (c) Now argue that
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