# I don't know what operation to preform with “!”

I am currently working on my math homework (the book was optional), and I don't know what to do when I see the "!". I am trying to evaluate a problem with this symbol.

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For future reference: List of mathematical symbols – Rahul Nov 28 '12 at 2:42

Assuming it isn't an exclamation point, $!$ is a factorial. That is for a positive integer $n$

$$n!=n(n-1)(n-2)\cdots 2 \cdot 1.$$

For instance $5!=5\cdot4\cdot3\cdot2\cdot1=120$.

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There are several rather common operations associated with $!$. By the far the most common is the factorial which is defined for natural numbers as $$n! = n\times (n-1)!\ \ \ \text{with}\ \ \ 0! = 1$$ This is simply a product of all the natural numbers from $1$ to $n$. Incidentally, this is also the number of ways to rearrange $n$ distinct objects into order.

For your interest, another commonly used and related notation is the double factorial $n!!$ This is often used to denote the product of the odd numbers from $1$ to $n$.

Yet another one which uses an exclamation mark (although this one is much rarer from my experience) that I have seen is the number of derangements or the subfactorial. This is sometimes denoted $!n$ and is given as $$!n = n\times !(n-1) + (-1)^n\ \ \text{with}\ \ \ !0 = 1$$

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n!=n×(n−1)! I don't quite understand this. I got the answer 20, when n = 5. – Shane Kelsey Nov 28 '12 at 2:51
You have $n! = n \times (n-1)!$ Note the exclamation mark at the end of the $(n-1)$. You are repeatedly applying the definition, it is not $n! = n \times (n-1)$ without the $!$ – EuYu Nov 28 '12 at 2:54
It seems recursive. I was a bit intimidated by that. – Shane Kelsey Nov 28 '12 at 2:55
Well it is recursively defined. If it helps, we don't really apply the recursive definition in most cases. We simply understand it to be the product of all the integers between $1$ and $n$. – EuYu Nov 28 '12 at 2:56

You may want to prove that the number of different subsets with $\,k\,$ elements that a set with $\,n\,$ elements , $\,0\leq k\leq n\,$ , is given by what's called **the binomial coefficient\binom{n}{k}:\stackrel{\text{Def.}}=\frac{n!}{k!(n-k)!}

If the above is way too hard, you can try to work out different cases: for any $\,n\,$, the cases $\,k=0,1,2,,n-2,n-1,n\,$ . On purpose I recommend you these ones as you will find something really nice, and sometimes surprising at the beginning, about these cases. Perhaps first choosing, say $\,n=3,4\,$ , then something more general.

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A $:=$ with Def over it, perhaps the least ambiguous use of notation I've ever seen. – JSchlather Nov 28 '12 at 2:53
That's usually less cumbersome and less long than explaining to begining students that := in mathematics means "LHS defined by RHS". – DonAntonio Nov 28 '12 at 3:11