Arrangements with identical elements

Count arrangements of objects where some are identical by dividing by the arrangements of each identical group, including words with repeated letters and patterns of two kinds of object.

Dividing out the repeats

The letters of TOOT give only $6$ different words, not $4! = 24$, because swapping the two Ts, or the two Os, gives the same word. Each word was counted $2! \times 2! = 4$ times.

Example 1●

How many arrangements are there of the letters of BANANA?

$\dfrac{6!}{3!\,2!} = \dfrac{720}{12} = 60$

6 letters: three As, two Ns, one B.

Two kinds of object

Example 2●●

Six women and four men queue at a bus stop. How many patterns of men and women are possible?

$\dfrac{10!}{6!\,4!} = 210$

Only the pattern matters, so the women are treated as identical, and so are the men.

With restrictions

Example 3●●●

How many arrangements of SOCKS have the two Ss together?

Arrange SS, O, C, K: $4! = 24$.

Glue SS into one block. The block's two letters are identical, so it has only 1 internal order.

So $60 - 24 = 36$ arrangements have the Ss apart.

All arrangements: 5!/2! = 60.

Flashcards

Arrangements with repeats

$\dfrac{n!}{a!\,b!\cdots}$

$r$ Ys and $n - r$ Ns

$\dfrac{n!}{r!(n - r)!}$

Block of identical letters

Only $1$ internal arrangement.

Practice

How many arrangements are there of BANANA?
How many arrangements are there of TOOT?
Six women and four men queue. How many patterns of men and women are possible?
In a queue of ten adults, how many patterns of men and women are possible (any numbers of each)?
How many arrangements are there of SOCKS?
How many arrangements of SOCKS have the two Ss together?
How many arrangements are there of MISSISSIPPI?
The digits $1, 1, 1, 2, 2, 3$ form six-digit numbers. How many are possible?
Eight balls, $7$ red and $1$ white, are lined up. The number of arrangements is: