Factorials, Permutations, and Combinations

The first few factorials are 1 = 1, 2 = 2, 3 = 6, 4 = 24, 5 = 120. As you can see, the factorial function increases rather steeply. Our original example involved 12 = 479,001,600. The symbol for factorial is the same as an exclamation point. (Thus, be careful when you write something like, To our amazement, the membership grew by 1001 ) 

Can factorials also be defined for nonintegers Later, we will introduce the gamma function, which is a generalization of the factorial. Until then you can savor the amazing result that 

Our first example established a fundamental result in combinational algebra the number of ways of arranging n distinguishable objects (e.g., with different colors) in n different boxes equals n . Stated another way, the number of possible permutations of n distinguishable objects equals . 

Suppose you have n good friends seated around a diimer table who wish to toast one another by clinking wineglasses. How many clinks will you hear The answer is the number possible combinations of objects taken 2 at a time 

You can also deduce this more directly by the following argument. Each of n diners clinks wineglasses with his or her n—1 companions. You might first think there must be n(n — 1) clinks. But, if you listen carefully, you will realize that this counts each clink twice, one for each clinkee. Thus, dividing by 2 gives the correct result n(n — l)/2. 

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