Chapter 8.4, Problem 49MS

Keep on cancellin’! Building on the previous Mindscape, notice that the numerator of the given formula, $T\times (T-1)\times (T-2)\times \mathrm{...}\times (T-(S-1)$ counts the number of ways to order or arrange S objects taken from a total of Tobjects. That is, there are Tchoices for the first object, $T-1$ choices for the second object, $T-2$ choices for the third object, and so on, until S objects have been chosen. (Note that there are S factors in the formula, one for each choice.) Verifythat $T\times (T-1)\times (T-2)\times \mathrm{...}\times (T-(S-1)=\frac{T!}{(T-S)!}.$