Permutation, Combination & Probability

The fundamental principle of counting; factorials; permutations with and without repetition, of letters with repeated letters, and round a circle; combinations and their standard properties; when order matters and when it does not; selections with at least or at most conditions; classical probability, the complement rule, addition and multiplication rules, mutually exclusive versus independent events; dice, cards, coins and drawing balls — with fully worked examples.

Permutation, Combination & Probability

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📑 Contents (6 sections)

Last reviewed 22 Sept 2026 · 5 min read

Counting first

FormulaFundamental principle

If one job can be done in $m$ ways and, independently, a second in $n$ ways, then both together can be done in $m \times n$ ways (AND → multiply). If a job can be done by either of two exclusive methods, in $m$ or in $n$ ways, the total is $m + n$ (OR → add).

$n! = n (n - 1) \dots 2 \cdot 1$ , with $0! = 1$ .

Permutations — order matters

FormulaArrangements

^{n} P_{r} = \frac{n !}{( n - r )!}

All $n$ objects: $n!$
With repetition allowed, $r$ places from $n$ types: $n^{r}$
$n$ letters of which one letter repeats $p$ times and another $q$ times: $\frac{n !}{p ! q !}$
Circular arrangements of $n$ objects: $(n - 1)!$ , and $\frac{( n - 1 )!}{2}$ when clockwise and anticlockwise are the same (a garland or necklace)
Objects that must stay together: tie them into one block, arrange the blocks, then arrange within the block

Combinations — order does not matter

FormulaSelections

^{n} C_{r} = \frac{n !}{r ! ( n - r )!},^{n} C_{r} =^{n} C_{n - r},^{n} C_{r} = \frac{^{n} P _{r}}{r !}

$^{n} C_{0} =^{n} C_{n} = 1$ ; $^{n} C_{1} = n$
$^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$
Total subsets of an $n$ -set: $2^{n}$
"At least one" is usually easiest as total $-$ none

A quick test: a committee is a combination (Ram and Shyam is the same committee as Shyam and Ram), while a president-and-secretary pair is a permutation.

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