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Factorial is not really a real function (see later), but it is so important that we include it here among useful functions.

Definition. Let k be a positive integer. We define the factorial of k as

k !=1·2·3·...·( k -1)· k .

We also define 0!=1.

Examples: 2!=1·2=2, 5!=1·2·3·4·5=120, 7!=5040.

Factorial grows very quickly, for instance 10!=3,628,800 and 20! is a number with 19 digits. 100! is so large that most calculators refuse o calculate it (usually the best they can do is 69!, which is about 1.7·10 98 ).

An important property of factorials is that they readily cancel in fractions. For instance:

Similarly

The down side is that there are no nice algebraic properties, for instance there are no nice formulas for things like (2 k )!, k k ! etc.

Such expressions cause mistakes when factoring, here is one correct example:

Nothing more can be done with it.

There is a thing called double factorial , where instead of multiplying all numbers one takes just every second.

Definition. Let k be a positive integer. We define k !! as k !!=1·3·5·...·( k -2)· k for k odd, k !!=2·4·6·...·( k -2)· k for k even.

Examples: 7!!=1·3·5·7=105, 10!!=2·4·6·8·10=3840. As usual, 0!!=1.

Factorial is a function whose domain is natural numbers and zero. Is it possible to extent this definition to a larger set (like to all non-negative real numbers) by somehow connecting the dots in the graph? This would be very helpful, for instance it would allow us to use the l'Hospital rule when calculating limits with factorials. The good new is that the answer is yes, there is a function defined on all real numbers apart from negative integers which agrees with factorial at natural numbers, see the Gamma function . The bad news is that this function is not exactly easy to work with and in particular, the tricks for limits that we covered are much better in dealing with factorials than messing around with Gamma functions.

Binomial coefficients

Definition. Let n k be non-negative integers. We define

This is called the binomial coefficient , also binomial number , or combinatorial number . We read this expression " n choose k ".

Example:

Some identities:

The last identity is very interesting, since on the right there are no factorials on the top, that is, the variable n is free to change. Thus we can introduce a useful generalization.

Definition. Let x be a real number, k a natural number. We define the binomial number by

Examples:

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Dan Shaffer grew up in Lancaster, Pennsylvania, and attended the Juniata College where he degrees in International Studies and German. Through Juniata, he studied abroad in Marburg, Germany where he studied Political Science. While at Juniata, Dan devoted much of his time to campus programming by serving as a Senior Resident Assistant, a Study Abroad Peer Advisor, Policy Chair for Juniata’s student government, and as the Content Coordinator for Juniata’s school newspaper, The New Juniatian . Dan is an Eagle Scout and recipient of the Eagles Abroad fellowship at Juniata College. In his free time, Dan likes learning and speaking foreign languages, playing computer games, writing/editing, listening to/creating music, hanging out with friends, and hiking. He is a part of the Northside cohort, and is serving with the Hill Community Development Corporation .

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