The idea

Permutations and combinations

Counting arrangements and selections — factorials, arrangements with repeated items, ordered selections (nPr), unordered selections (nCr), and the standard restriction tricks: objects kept together, objects separated, and "at least" counts built case by case or by complement.

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Statistics · Probability

Permutations and combinations

Why it works

Multiply the choices

n

different objects can be arranged in a line in

n! = n \times (n-1) \times \cdots \times 2 \times 1

ways:

n

choices for the first position,

n-1

for the second, and so on. That "multiply the choices" idea is the whole subject; everything else is bookkeeping.

Repeated objects

If some objects are identical, arrangements that only swap identical objects look the same, so the count shrinks. With

n

objects of which

p

are alike of one kind,

q

alike of another, …:

\frac{n!}{p!\,q!\cdots}.

For BANANA (

6

letters: three As, two Ns):

\dfrac{6!}{3!\,2!} = 60

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The rest of the explanation, plus 3 worked examples you step through move by move.

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