Solving Combination Problem: 11Cn=330

[temaire]

Homework Statement

Solve for n.
$$11_{C}_n=330$$

Homework Equations

$$n_{C}_r=\frac{n!}{(n-r)!r!}$$
Sorry if the combination formula looks bad. I don't know how to write the comb. formula with Latex.

The Attempt at a Solution

I solved for n as far as $$n!(11-n)!=133056$$ How do I go further with this? The answers are 4 or 7.

 

[Tedjn]

Honestly the easiest way to do this would be a combination of algebra and guess and check. You should see that both n and 11-n would solve the equation, so finding one answer would give you the answer.

The usual way to reduce factorials is to try to keep as much of the factorial together as possible, which often let's you see patterns easier. For instance:

$$_{11}C_n = \frac{11!}{n!(11-n)!} = 330 \implies n!(11-n)! = \frac{11!}{2 \cdot 3 \cdot 5 \cdot 11}$$

You can cancel out the 11 and the 10 on top, leaving you with:

$$n!(11-n)! = 3 \cdot 8!$$

Now you start with the guess and check. You know that n has to be less than 8.