I have created 25 watercolor panels, each a different color, and they hang in a grid that is 5 panels across and 5 panels down. How many different configurations and/or sequences of the panels are possible - even if I change the position of just 2 panels? The 5 x 5 grid configuration must be maintained.
I know the answer has a fairly simple equation - but I'm an artist out of math practice - and was curious as to the number of possible ways I can install this piece.
Thank you!
If we number the positions and start placing the pieces, the first piece has $25$ places, the second $24$, the third $23$ and so on, so in total we have
$$25\cdot 24\cdot 23\cdots 3\cdot 2\cdot 1$$
possibilities. The product of the numbers $1$ to $n$ is well known in mathematics, it is called the factorial of $n$ and it is denoted with $n!$. Here, we have $25!$ possibilities , which is equal to $$15,511,210,043,330,985,984,000,000$$