How to divide large factorial numbers?

Question

In my discrete mathmatics class, I am tasked with dividing a large factorial number by another large factorial number. The numbers are too big to use calculators. How can I do this problem?

$$\frac{640!}{20!\,620!} = ?$$

Answer 1

It's easier to see how this works with smaller numbers. Consider $\frac{10!}{7!\,3!}$. We can cancel like this:

$$\begin{align} \frac{10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{(7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1)(3\cdot 2\cdot 1)} &= \frac{10\cdot 9\cdot 8\cdot \not7\cdot \not6\cdot \not5\cdot \not4\cdot \not3\cdot \not2\cdot \not1}{(\not7\cdot \not6\cdot \not5\cdot \not4\cdot \not3\cdot \not2\cdot \not1)(3\cdot 2\cdot 1)}\\ &= \frac{10 \cdot9\cdot8}{3\cdot2\cdot1} \end{align}$$

This last expression is much easier to deal with. We can carry out more cancellations as we find them, and end up with an integer at the end. A more compact way of writing this out is this:

$$\frac{10!}{7!\,3!}=\frac{10\cdot9\cdot8\cdot7!}{7!\,3!}=\frac{10\cdot9\cdot8\cdot\not{7!}}{\not{7!}\,3!}=\frac{10\cdot9\cdot8}{3!}$$

In your case, we can write:

$$\frac{640!}{20!\,620!}=\frac{640\cdot 639\cdots621\cdot620!}{20!\,620!}=\frac{640\cdot 639\cdots621}{20!}$$

You can keep canceling from here, as each number on the bottom has a multiple on top, or you could probably use your calculator at this step. The number $20!$ is not too big for most calculators to handle, at $19$ digits. The numerator is $56$ digits, which is getting bit unwieldy... Cancellation is better.

Answer 2

This is the binomial coefficient $$\binom{640}{20}.$$ It equals $$\frac{640\times639\times638\times\cdots\times623\times622\times621}{20\times19\times18\times\cdots\times3\times2\times1}$$ and one can cancel everything in the denominator from the numerator. It will leave a reasonably practical multiplication. I'd prefer not to have to actually do it. $\ddot\smile$

Answer 3

You can use the calculator to do all the simplifications but the get the result you must do something by hand because the result is anyway too big to be memorized by a calculator, $38$ digits $$40\,477\,522\,598\,018\,531\,284\,007\,399\,717\,734\,668\,000$$

I have simplified for you

$$\small\frac{640\cdot 639\cdot 638\cdot 637\cdot 636\cdot 635\cdot 634\cdot 633\cdot 632\cdot 631\cdot 630\cdot 629\cdot 628\cdot \ 627\cdot 626\cdot 625\cdot 624\cdot 623\cdot 622\cdot 621}{{20\cdot 19\cdot 18\cdot 17\cdot 16\cdot 15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2}}$$

you need just to compute with a calculator

$$a = 23\times 311\times 89\times 13\times 313\times 7\times 10\\ b = 33\times 157\times 37\times 7\times 631\times 79\times 71\times 10\\ c = 633\times 634\times 127\times 106\times 58\times 15$$ and finally by hand $$a\cdot b\cdot c=40\,477\,522\,598\,018\,531\,284\,007\,399\,717\,734\,668\,000$$

Hope this can help