factorial division when the bottom number is larger than the top number

Question

I have a factorials problem to solve, and I do not know the method of solving it.

I know how to do one number factorials (e.g. 5!, 15! etc...) and factorial division where the top number is larger than the bottom number.

Could someone please make a simple guide (with an example question/answer please) on how to solve a factorial divison such as:

100!/102!

I've tried so many searches on google no one has expained this type of question

Thank you very much.

Answer 1

Hint:

$$\dfrac{100!}{(102)!} = \dfrac{100!}{102\cdot 101\cdot 100!}$$

Cancel.

Answer 2

Let's make the numbers a little smaller and write it out:

$$\frac{10!}{12!} = \frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}.$$

All but the $12$ and $11$ in the denominator cancel. But this can be written to make this evident:

$$\frac{10!}{12!} = \frac{10!}{12 \cdot 11 \cdot 10!} = \frac{1}{132}.$$

You're still canceling all of the numbers you did in the first expression, but the notation is much more compact.

Now apply this to your problem:

$$\frac{100!}{102!} = \frac{100!}{102 \cdot 101 \cdot 100!} = \frac{1}{10302}.$$

Answer 3

If you know how to handle things when the big guy is on top, start from

$\frac{100!}{102!}$;

Flip it over: We get

$\frac{102!}{100!}$;

Simplify, which you know how to do. We get

$(101)(102)$;

Flip it over again: We get

$\frac{1}{(101)(102)}$, and now we are finished.

After a couple of times, you will be able to skip the middle steps and go immediately to the answer.