Question

Can someone please help me simplify x(x-1)(x-2)(x-3)...(x-n)?

Answer 1

Answer:

Observe that the coefficient of x1 in

(x−1)(x−2)(x−3)

is

2⋅3+1⋅2+1⋅3

or the coefficient of x4−2 in

(x−1)(x−2)(x−3)(x−4)=(x−1)(x−4)(x−2)(x−3)=(x2−(1+4)x+1⋅4)(x2−(2+3)x+2⋅3)

is

2⋅3+(1+4)(2+3)+1⋅4

So, the required sum

=∑1≤r1<r2≤nr1r2=(∑nr=1r)2−∑nr=1r22