Let a1,a2....,an be a random permutation of [1,2,...n] where n is an odd natural number.

Show that the product

P = (a₁-1)*(a₂-2)*...*(a_n-n)

is an even integer.

proof:

we know (a1+ ..+ an) = ( 1+ 2...+n) as LHS is a pemuation of 1 to n

so (a1-1)+ (a2- 2) + ... + ( an-n)= 0

now there are odd number of terms on the left and all the terms on left cannot be odd as sum is 0 that is even

so one the terms has to be even

hence product is even. Done

taken from puzzle-a-day.blogspot.in/2010/10/even-product.html