Olympiad workbookIOQMPermutation and Combination
A positive integer k is said to be good if there exists a partition of 1,2,3, , 20 in to disjoint proper subsets such that the sum of the numbers in each subset of the partition is k. How many good numbers are there?
Correct answer
6
Step-by-step solution
Sum of numbers equals to 20 21 2 =210 & 210=2 3 5 7 Number of Partition Sum I 2 105 II 3 70 III 5 42 IV 7 30 V 6 35 VI 10 21 So K can be 21,30,35,47,70,105 Good numbers equal to 6 Case-I: A= 1,2,3,4,5,16,17,18,19,20 , B= 6,7,8,9,10,11,12,13,14,15 Case-II: A = 20,19,18,13 , B = 17,16,15,12,10 , C = 1,2,3,4,5,6,7,8,9,11,14 Case-III: A= 20,10,12 , B= 18,11,13 , C= 16,15,9,2 , D= 19,8,7,5,3 , E= 1,4,6,14,17 Case-IV: aligned A & = 20,10 , B= 19,11 , C= 18,12 , D= 17,13 , E= 16,14 , F= 1,15,5 , G & = 2,3,4,6,7,8 aligned