Combinations Calculator can calculate the number of ways to select r outcomes from n possibilities when the order of the items chosen in the subset does not matter.

What is the "Combinations"?

In mathematics, "Combinations" refer to the ways of selecting items from a larger set where the order of selection does not matter. In other words, combinations focus on the selection of elements without regard to the sequence in which they are chosen.

The number of combinations is usually denoted by the symbol $\binom{n}{r}$, read as "n choose r." It represents the number of ways to choose r elements from a set of n elements. The formula for calculating combinations is:

$\binom{n}{r} = \frac{n!}{r!(n - r)!}$

where
$n!$ (n factorial) is the product of all positive integers up to n.

For example:

• If you have 5 different fruits (apple, banana, orange, grape, and pear) and you want to choose 3 out of them, the number of combinations is $\left(\genfrac{}{}{0px}{}{5}{3}\right)=\frac{5!}{3!(5-3)!}=\frac{5\times 4\times 3\times 2\times 1}{3\times 2\times 1\times 2\times 1}=\mathrm{10<span\; style="font-family:宋体;"><br></span>}$

• This means there are 10 different ways to choose 3 fruits out of the 5.

Combinations are particularly important in probability and statistics, where they are used to calculate the likelihood of different outcomes.