If the sum of the products of all possible pairs of distinct numbers chosen from the set 1, 2, 3, , n is 546 , then the value of n is

Step-by-step solution

Let S be the sum of the products of all possible pairs of distinct numbers from the set 1, 2, 3, , n . We know the identity: ( _ k=1 ^ n k )^2 = _ k=1 ^ n k^2 + 2S Therefore, S = 1 2 [ ( _ k=1 ^ n k )^2 - _ k=1 ^ n k^2 ] Substituting the standard summation formulas: S = 1 2 [ ( n(n+1) 2 )^2 - n(n+1)(2n+1) 6 ] S = n(n+1) 4 [ n(n+1) 2 - 2n+1 3 ] S = n(n+1) 4 [ 3n^2 + 3n - 4n - 2 6 ] S = n(n+1)(3n^2 - n - 2) 24 Factoring the quadratic term 3n^2 - n - 2 = (n-1)(3n+2) gives: S = (n-1)n(n+1)(3n+2) 24 Given S = 546 , we h