Question

9. Give a proof of the following identity using a double-counting argument: -k) = (m+n) Then using this result, derive the following special case from it. This can be done algebraically in just a few steps (you don't need to give a separate counting argument for this): Ëm*- (1)

Answer 1

Sol? :s claims Ž (26" M) = (mama) - Cro Right hand size of Ø gives number of ways to select a subset o f 8 people from set of man people Now we divide a man people into two groups. Group © contains m people of Group @ contains semainig n people case :) For selection of a people first we select o people from first group and r people from second group Number of selection = 11

group 4 case ii) Now select a person som first f (rond) people from second group : Number of selection - (2009.) case (1) Now select a person from post group for al people nom second group. .: Number of selection (m) (82) - continuing in this way. case sy Now select r person som first group f o people from second group. Number of selections (m2(?). . : Number of way to select r people from of man people set (O)()+(1269.) +---+ (m) :) (man) = E(F) (22) fare particular no moman & retran KSO Ź (2) k) = 100) Ź (396) --- ()-122 C / ICO