I have solved it as per my knowledge and understanding. Since there are 4 elements so no. of permutations will be $$4! = 24$$ $$(1,2,3,4), (1,2,4,3), (1,3,2,4), (1,3,4,2), (1,4,2,3), (1,4,3,2), (2,1,3,4), (2,1,4,3), (2,3,1,4), (2,3,4,1), (2,4,1,3), (2,4,3,1), (3,2,1,4), (3,2,4,1), (3,1,2,4), (3,1,4,2), (3,4,2,1), (3,4,1,2), (4,2,3,1), (4,2,1,3), (4,3,2,1), (4,3,1,2), (4,1,2,3), (4,1,3,2)$$

Is it correct?

• And for even and odd-

Even: product of even no of transpositions. e.g., $(1,2,3) = (1,2)(1,3)$ is even Odd: product of odd no of transpositions. e.g., $(1,2,3,4) = (1,2)(1,3)(1,4)$ is odd.

• But if I apply this theory in this question then all permutations will be odd. I am confused here, is my solution incorrect? May be there will be more permutations with less elements like $(1,2,3) , (1,2,4)$?

• Also there is a swapping logic! If the numbers are swapped odd times then it is odd and even otherwise

So in this case $(1,2,3,4)$ is even (no swaps) $(3,2,1,4)$ = Swap 1<->3 is odd Also $(4,2,1,3)$ = Swap 3<->4 then swap 1<->4 = even permutation

What will be the answer to my question?