Arrangements with repetition and restrictions

10 minAS Level5.2.2

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can count the arrangements of objects in a line when some are identical, and when there are rules like "must be together", "must be apart" or "must be in certain places".

Key to the symbols

Symbol	Meaning
$n!$	" $n$ factorial": $n \times (n - 1) \times \cdots \times 2 \times 1$ . For example $4! = 4 \times 3 \times 2 \times 1 = 24$
$0!$	defined to be $1$
${}^n C_r$	the number of ways to choose $r$ things from $n$ when order doesn't matter: $\dfrac{n!}{r!\,(n - r)!}$

Why $n!$ ? To arrange $n$ different objects in a line, there are $n$ choices for the first place, then $n - 1$ for the second, and so on down to 1. Multiply them: $n!$ .

When some objects are identical

Arrange the letters A, A, B. If the two A's were different (call them A₁ and A₂), there would be $3! = 6$ arrangements.

But A₁A₂B and A₂A₁B look the same: both are AAB. Every real arrangement has been counted $2!$ times, once for each way of ordering the A's.

So divide: $\dfrac{3!}{2!} = 3$ . They are AAB, ABA, BAA.

Definition

Arrangements with identical objects

$n$ objects, with $n_1$ identical of one kind, $n_2$ identical of another kind, and so on, can be arranged in a line in

$\frac{n!}{n_1!\,n_2!\,\cdots}$

ways. (Objects that appear only once contribute $1! = 1$ , so they can be left out.)

Worked example: Example 1: BANANA

How many different arrangements are there of the letters of BANANA?

Have a go first?

Quick check

CalculateQuestion 1 · Apply

Answer the question above to continue.