Caution

This article defines permutations and combinations in terms of factorials. When solving problems, sometimes $0!$ (zero factorial) makes an appearance. Please keep in mind that $0! = 1$ , not zero.

Permutations

A permutation is an ordered arrangement of distinct items. For example, permutations of all four of the letters ABCD include DBAC, CDAB, ABCD, and more. We can also create permutations of only three of the letters, such as ABC, DAB, BAD, or CAD.

The number of ways to permute $n$ out of $m$ total items is written as $_{m} P_{n}$ and can be computed using either an explicit formula involving factorials or built-in calculator functionality.

Explicit formula for permutations

$_{m} P_{n} = \frac{m !}{( m - n )!}$

Computing permutations with Desmos

In the Desmos Scientific Calculator and the Desmos Graphing Calculator, permutations are computed via the $nPr$ function, which takes two inputs. For example, to compute $_{7} P_{3}$ , type nPr(7,3). The calculator will then display $nPr (7, 3) = 210$ .

Computing permutations with the TI83/TI84

In the main calculator input window, input your $n$ value.

Click the MATH button and use the → key to switch to the PROB (probability) menu. Then, select nPr.

Type your $r$ value on the screen. For example, if you are looking to compute $_{8} P_{5}$ , you should now see 8 nPr 5 on your calculator screen.

Press ENTER to evaluate.

Other miscellaneous notes:

$r$ must be less than or equal to $n$
$_{n} P_{0} = 1$
$_{n} P_{1} = n$
$_{n} P_{n} = n!$

Combinations

A combination is an unordered arrangement of distinct items. For example, going back to our set of four of the letters ABCD the three-letter combinations BAD and DAB would be considered to be the same combination since they contain exactly the same letters.

The number of ways to combine $n$ out of $m$ total items is written as $_{m} C_{n}$ or as $(n m)$ and can be computed using either an explicit formula using factorials or built-in calculator functionality. In contexts where the notation $(n m)$ is used, the term binomial coefficient is also used, which stems from its relationship with the binomial theorem.

Explicit formula for combinations (a.k.a. - binomial coefficient)

$(n m) =_{m} C_{n} = \frac{m !}{( m - n )! n !}$

Note that by this formula we can establish the identity $_{m} P_{n} = n! \cdot_{m} C_{n},$ which also implies that $_{m} P_{n} \geq_{m} C_{n},$ meaning that the number of combinations cannot exceed the number of permutations.

Computing combinations with Desmos and the TI83/TI84

Follow the same instructions as for permutations described above, but instead use the $nCr$ function.

Other miscellaneous notes:

$r$ must be less than or equal to $n$
$_{n} C_{0} = 1$ and $_{n} C_{n} = 1$
$_{n} C_{1} = n$ and $_{n} C_{n - 1} = n$
$_{n} C_{0} +_{n} C_{1} +_{n} C_{2} + \dots +_{n} C_{n} = 2^{n}$

Combinations and Pascal’s Triangle

There is a relationship between the combination numbers and the entries of Pascal’s triangle, as shown below. Corresponding entries in each triangle are equal in value to each other.

11112113311464115101051 ⋮_{0} C_{0}_{1} C_{0}_{1} C_{1}_{2} C_{0}_{2} C_{1}_{2} C_{2}_{3} C_{0}_{3} C_{1}_{3} C_{2}_{3} C_{3}_{4} C_{0}_{4} C_{1}_{4} C_{2}_{4} C_{3}_{4} C_{4}_{5} C_{0}_{5} C_{1}_{5} C_{2}_{5} C_{3}_{5} C_{4}_{5} C_{5} ⋮ (0 0) (0 1) (1 1) (0 2) (1 2) (2 2) (0 3) (1 3) (2 3) (3 3) (0 4) (1 4) (2 4) (3 4) (4 4) (0 5) (1 5) (2 5) (3 5) (4 5) (5 5) ⋮

Trevor's Notes

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Combinations and Permutations

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Permutations

Combinations

Combinations and Pascal’s Triangle

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