### Instruction: Determine the number of elements of order 2 in the symmetric group S_4.  Answer Choices: (A) 1 (B) 3 (C) 6 (D) 9.### Response: An element in the symmetric group S_4 has order 2 if and only if it can be written as a product of disjoint 2-cycles. Let's check each option: (A) 1: This is incorrect. There is more than one element of order 2 in S_4. (B) 3: This is incorrect. There is more than three elements of order 2 in S_4. (C) 6: This is incorrect. There is more than six elements of order 2 in S_4. (D) 9: This is correct. The elements of order 2 in S_4 are exactly the 2-cycles and the products of two disjoint 2-cycles. There are 6 2-cycles and 3 products of two disjoint 2-cycles, giving a total of 9 elements of order 2. The answer is D.