binomial distribution formula - factorial cancellation

Question

This is confusing me in Intro to Stats course.. any help in explaining is appreciated!

Binomial Distribution formula: $\frac{n!}{(n-k)!k!}$

For 125 Coin Flips w/ 3 Heads = 317,750 (n = 125, k = 3)

Steps to solve:

1. $\frac{125!}{(125-3)!3!}$
2. $\frac{125\bullet124\bullet123}{3\bullet2\bullet1}$
3. $\frac{1,906,500}{6}$ = 317,750

In the steps to arrive at this answer explain that in step 2 125! cancels out (125 - 3)!. I do not understand why ($125\bullet124\bullet123$) cancels ($122\bullet121\bullet120$).

Thanks in advance - I hope I articulated my question well enough to get an answer!

Answer 1

You're misinterpreting $(n-k)!$. What you have is

1. $125! = 125 \cdot 124 \cdot 123 \cdot \ldots \cdot 3 \cdot 2 \cdot 1$

2. $(125 - 3)! = 122! = 122 \cdot 121 \cdot 120 \cdot \ldots \cdot 3 \cdot 2 \cdot 1$

As you can see, in the first case, you have the product of all positive integers from 1 to 125 (inclusively) and in the second case, all positive integers from 1 to 122. When you take the ratio of the first quantity to the second, you can match all positive integers from 1 to 122, so you are left with $125 \cdot 124 \cdot 123$ in the numerator.