There are 8 distinct weights of 1 kg , 2 kg , 3 kg , 4 kg , 5 kg , 6 kg , 7 kg , and 8 kg . A balance scale has 8 distinct numbered slots, with slots 1 to 4 on the left pan and slots 5 to 8 on the right pan. The number of ways to place all 8 weights into these 8 slots, such that each slot receives exactly one weight and the total weight on the left pan is exactly 4 kg more than the total weight on the right pan, is

Step-by-step solution

Let S_L and S_R be the sum of the weights on the left and right pans, respectively. We are given S_L - S_R = 4 . The total sum of all 8 weights is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 . Thus, S_L + S_R = 36 . Solving these two equations gives 2S_L = 40 S_L = 20 and S_R = 16 . We need to select 4 weights for the right pan such that their sum is 16 . The 4 -element subsets of 1, 2, 3, 4, 5, 6, 7, 8 that sum to 16 are: 1, 2, 5, 8 1, 2, 6, 7 1, 3, 4, 8 1, 3, 5, 7 1, 4, 5, 6 2, 3, 4, 7 2, 3, 5, 6 There are exactly 7 such