All courses › Mathematics R1 › Ordered and unordered selections

Ordered and unordered selections

When order matters, you multiply the number of choices for each place: first $n$ , then $n - 1$ , and so on. When order does not matter, you divide by the number of ways the chosen elements can be arranged. That is the binomial coefficient. A committee is unordered, while podium places are ordered. The division removes exactly the orders you do not want to count.

Try it yourself

The binomial distribution

\binom{n}{k} = \frac{n!}{k!(n-k)!}

unordered selection

n(n-1)\cdots(n-k+1)

ordered without replacement

Symbols

$n$	the number to choose from
$k$	the number you choose

Example

$\binom{5}{2} = \dfrac{5 \cdot 4}{2 \cdot 1} = 10$ . Gold, silver and bronze among 8 runners can be awarded in $8 \cdot 7 \cdot 6 = 336$ ways.

First ask whether the order matters. That decides whether you should divide by

k!

Practise probability and combinatorics for free →

← Perpendicular vectors · Binomial probability →

Part of Mathematics R1: Probability and combinatorics.