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12. Consider the following combinatorial identity: n. n k : . 2911 2 (k) n r: = 1 a. Present a combinatorial argument for this identity by considering a set of n 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 committee of size it and its chairperson? ii. How many possible selections are there of a chairperson and the other committee members? b. Verify the following identity for n = 1, 2, 3, 4, 5: Z (1:) k2 = 2"_2n{n + 1) k = 1 For a combinatorial proof ofthe preceding, consider a set of 11 people and argue that both sides of the identity represent 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 committee containing exactly it people? ii. How many different selections are there in which the chairperson and the secretary are the same? (ANSWER: 112"" 1. ) iii. How many different selections result in the chairperson and the secretary being different? c. Now argue that 1'1 2 (D k3 = 2\"'3n2(n + 3) k = 1