GATE 2020 MA – Question 26
Find the number of elements of order 3 in the symmetric group S₆.
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Correct answer: 80 to 80
Explanation
An order-three permutation in S₆ has cycle type (3,1,1,1) or (3,3). Single 3-cycles number $\binom63(3-1)!=20(2)=40$. Two disjoint 3-cycles number $6!/(3^2\,2!)=40$. The identity is excluded. Thus there are 40 + 40 = 80 elements of order three.