What does ${50}\choose{4}$ mean in statistics?

Question

I have a test tomorrow in statistics and was wondering what the following means?

$$\binom{50}{4}$$

My professor along with most of my classmates have a calculator they can just plug that into. The professor never went into depth on what it means and how to figure it out with a normal ti-30.

Thanks for your time.

Answer 1

${50}\choose{4}$ is read "$50$ choose $4$" and is the number of ways to choose $4$ things from $50$ things, where order doesn't matter. Most graphing calculators will have a button for this, if not there is the formula:

$${{n}\choose{k}} =\frac{n!}{k!(n-k)!}$$

where $n!=n\times (n-1)\times\cdots\times 1$.

Answer 2

Binomial Coefficient or Combination

Wikipedia words it quite nicely:

$\binom{n}{k}$ is read as "$n$ choose $k$", because there are $\binom{n}{k}$ ways to choose $k$ elements, disregarding their order, from a set of $n$ elements.

I'd bet that most calculators use this efficient method, $$\binom{n}{k} = \prod\limits_{i=1}^k \frac{n+1-i}{i}$$ So for your example, we have $$\binom{50}{4} = \prod\limits_{i=1}^4 \frac{51-i}{i}=\frac{50\cdot 49\cdot 48\cdot 47}{1\cdot 2\cdot 3\cdot 4}=230,300$$ Which means that there are 230,300 ways to choose 4 elements from a set of 50 elements, disregarding their order. The more common but less efficient method of computing this is $$\binom{n}{k} = \frac{n!}{k!(n-k)!}\quad\mbox{for}\quad 0\leq k\leq n$$ So for your example, we have $$\binom{50}{4} = \frac{50!}{4!46!}=\frac{50\cdot 49\cdot 48\cdot 47\cdot 46!}{1\cdot 2\cdot 3\cdot 4\cdot 46!}=230,300$$ Have a look at Binomial Coefficient, Combination and Factorial for a better understanding.