Chapter 8.4, Problem 48MS

Cancellation! (H) This section explains that the number of ways to select S objects from a total of T objects equals

$\frac{T\times (T-1)\times (T-2)\times \mathrm{...}\times (T-(S-1)}{(S\times (S-1)\times (S-2)\times \mathrm{...}\times 2\times 1)}.$

The denominator of this formula is the product of a natural number S and all the natural numbers less than S. Mathematicians denote this product S!, read “S factorial.” So, for example, $3!=3\times 2\times 1=6,5!=5\times 4\times 3\times 2\times 1=120,$ and $1!=1.$

Verify that

$\frac{T\times (T-1)\times (T-2)\times \mathrm{...}\times (T-(S-1)}{(S\times (S-1)\times (S-2)\times \mathrm{...}\times 2\times 1)}=\frac{T!}{(T-S)!}.$