October 2026
It might be helpful to write out in words what \(P(4,1)\), \(P(4,2)\), \(P(4,3)\), and so on, actually mean.
There are many ways to approach this. It might be useful to calculate \(P(n,1)\), \(P(n,2)\), \(P(n,3)\), and so on, for some specific values of \(n\) (in (a), this was done for \(n=4\)). To tidy up your expression, it might also be useful to read about factorials and binomial coefficients.
Using your formula from (b), consider the quantity \(\dfrac{P(n,k+1)}{P(n,k)}\).
Use part (c) and look for a pattern.