question on set theory
Please help me in solving the following question How many nonempty subsets of {1, 2, 3 . . . 12 } have the property that the sum of the largest element and the smallest element is 13?
Let the smallest number be 1
Let the smallest number be 1 then the largest number will be 12. Now we have to find all the possible subsets with above two numbers and including/excluding numbers from 2 to 11(10 numbers). If we take the smallest number as 2 then the largest number will be 11. Here the subsets can contain numbers from 3 to 10 with above two numbers (8 numbers). Let be more generic and consider the smallest number to be n then largest number will be 13-n. The remaining elements of the sub set can chosen from 12 –2n different elements. n + 1, n + 2 , . . . (13 - n) Thus the total number of possible sets are 1212-2n And n ≤ 7 as anything more than n=6 will have duplicate values. So the answer is 210 + 28+ . . . +20 = 1365