Why does $(n-1)(n−2)\cdots(n−)+(n−1)(−2)\cdots(−+1)= (−1)(−2)\cdots(−+1)(−+)$?

Question

Why is it equal? $$(n-1)(n-2)...(n-k)+k(n-1)(n-2)...(n-k+1)=(n-1)(n-2)...(n-k+1)(n-k+k)?$$

I tried common factor and algebra tricks.

What is done here?

I have tried common factor but it doesn't help.

Answer 1

\begin{align} LHS & = \pmb{(n-1)(n-2)...(n-k+1)}(n-k) + k\pmb{(n-1)(n-2)...(n-k+1)} \\ & = \pmb{(n-1)(n-2)...(n-k+1)}[(n-k)+k] \\ & = (n-1)(n-2)...(n-k+1)(n-k+k) \\ & = RHS \end{align}

The bold part is the common factor.