Suppose you have a list of integers from 1 to N
And say you remove any two numbers, X and Y, from the list of numbers, and add (X + Y + X.Y) to the list
If you keep doing this until you have only one number remaining (which means I performed the above operation N-1 times), the answer turns out to be (N+1)! - 1
For example, let N = 4 Which means my list is: {1, 2, 3, 4}
If I choose X = 1 and Y = 2, then (1 + 2 + 1.2) is 5, so my list now becomes {5, 3, 4}
Now let X = 5 and Y = 4, then (5 + 4 + 5.4) is 29, so my list now becomes {29, 3}
And when X = 29 and Y = 3, we get (29 + 3 + 29.3) which is {119}
And 119 turns out to be 5! - 1 So for N = 4, my answer is 5!-1
Similarly for any N, my result ends up being (N+1)! - 1
Can someone provide some proof or explanation for this?
For reference, my question is based on this problem- https://www.codechef.com/MAY19B/problems/REDONE
Suppose your list is $L$. Then consider $I=\prod\limits_{i\in L}(i+1)$. $I$ is invariant, since when you do one of your operations, $x,y\to x+y+xy$, $(x+1)(y+1)=x+y+xy+1$ so $L$ remains constant. Thus $L_{\text{start}}=L_{\text{end}}$, so since $L_{\text{start}}=(n+1)!$, we must have $L_{\text{end}}=(n+1)!$ But there's just one number left so this number must be $(n+1)!-1$.