Let S1 = {(i, j, k): i, j, k are elements of {1, 2,..., 10}}, S2 = {(i, j): 1

Question

Let S1 = {(i, j, k): i, j, k are elements of {1, 2,..., 10}}, S2 = {(i, j): 1 <= i < j + 2 <= 10, with i, j in {1, 2,..., 10}}, S3 = {(i, j, k, l): 1 <= i < j < k < l, with i, j, k, l in {1, 2,..., 10}}, and S4 = {(i, j, k, l): i, j, k, l are distinct elements in {1, 2,..., 10}}. If n1, n2, n3, n4 denote the total number of elements in S1, S2, S3, S4 respectively, then which of the following statements are TRUE?

Accepted Answer

n1 = 1000. n1 = 10³ = 1000 (TRUE). n3 = C(10,4) = 210 (TRUE). n4 = 10*9*8*7 = 5040 (TRUE). For n2: condition is 1 <= i < j+2 <= 10, i.e., i <= j+1 and j <= 8; total ordered pairs (i,j) with i in 1..10, j in 1..10, i < j+2 means i <= j+1; count pairs where i,j in {1..10} and i - j <= 1, i.e., i <= j + 1. For each j from 1 to 10, i can range from 1 to min(j+1, 10). Also j <= 8 from j+2 <= 10. So j from 1 to 8: for j=1, i in {1,2}: 2 choices; j=2: i in {1,2,3}: 3;... j=8: i in {1,...,9}: 9. Wait, we also need i < j+2, meaning i <= j+1 and since i >= 1. Also j ranges from 1 to 8 (since j+2 <= 10). Total = sum_(j=1)⁸ min(j+1, 10) = sum_(j=1)⁸ (j+1) = 2+3+4+5+6+7+8+9 = 44. So n2 = 44 (TRUE).

Solution

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