Overview

Arrangement problems ask us to place objects in order.

Common examples include:

• Arranging letters to form words
• Arranging digits to form numbers
• Finding the rank of a word in dictionary order
• Finding the sum of all numbers formed from a set of digits

These problems combine permutations, repeated objects, and positional thinking.

The main idea is to count arrangements systematically rather than listing them.

Arranging Distinct Objects

If $n$ objects are all distinct, then the number of arrangements is $n!$.

Example 1

How many ways are there to arrange the letters in $MATH$?

The four letters are distinct.

So the number of arrangements is

Arranging Objects With Repetition

If some objects are repeated, then arranging them as if they were distinct overcounts.

Suppose there are $n$ total objects, with repeated groups of sizes

Then the number of distinct arrangements is

Example 2

How many distinct arrangements are there of the letters in $LEVEL$?

The word $LEVEL$ has $5$ letters.

The letter $L$ appears twice, and the letter $E$ appears twice.

So the number of distinct arrangements is

Restrictions in Arrangements

Many arrangement problems include restrictions.

Common restrictions include:

• Certain objects must be together
• Certain objects must not be together
• Vowels and consonants must alternate
• Digits cannot start with $0$
• Repeated objects are present

The usual strategy is to count the unrestricted arrangements first, then impose the restrictions carefully.

Objects Together

If several objects must stay together, treat them as a block.

Example 3

How many ways are there to arrange the letters $A,B,C,D,E$ if $A$ and $B$ must be together?

Treat $A$ and $B$ as one block.

Then we arrange the block together with $C,D,E$, giving $4$ objects:

Inside the block, $A$ and $B$ can be arranged in

ways.

So the total number of arrangements is

Objects Not Together

If certain objects must not be together, it is often easier to use complement counting.

Example 4

How many ways are there to arrange the letters $A,B,C,D,E$ if $A$ and $B$ must not be together?

There are

total arrangements.

From Example 3, the number where $A$ and $B$ are together is

Therefore, the number where they are not together is

Lexicographic Rank

The lexicographic rank of a word is its position when all arrangements are listed in dictionary order.

Usually, ranks start from $1$.

To find the rank of a word:

1. Look at the first position
2. Count how many valid arrangements begin with a smaller possible letter
3. Fix the actual first letter
4. Repeat for the next position
5. Add $1$ for the word itself

Worked Rank Example

Example 5

Find the rank of $MATH$ among all arrangements of the letters $A,H,M,T$ in alphabetical order.

The letters in alphabetical order are

We want the rank of

First consider the first letter.

Letters smaller than $M$ are

and

If the first letter is $A$, the remaining $3$ letters can be arranged in

ways.

If the first letter is $H$, the remaining $3$ letters can also be arranged in

ways.

So the number of words before $MATH$ with smaller first letter is

Now fix the first letter as $M$.

The second letter is $A$. There is no remaining letter smaller than $A$, so no new words are counted here.

Now fix

The remaining letters are

The third letter in $MATH$ is $T$. There is one smaller remaining letter, namely $H$.

If the third letter were $H$, the last letter would be forced. This gives

word before $MATH$.

So the total number of words before $MATH$ is

Therefore, the rank of $MATH$ is

Rank With Repeated Letters

When letters repeat, the same method works, but each count must account for repetition.

Example 6

How many arrangements of $LEVEL$ begin with a letter before $V$?

The letters before $V$ that appear in $LEVEL$ are

and

If the first letter is $E$, the remaining letters are

There are $4$ letters, with $L$ repeated twice, so the number of arrangements is

If the first letter is $L$, the remaining letters are

There are $4$ letters, with $E$ repeated twice, so the number of arrangements is

So the total number of arrangements beginning with a letter before $V$ is

Sum of All Arrangements of Digits

Some problems ask for the sum of all numbers formed by arranging given digits.

The key idea is positional contribution.

If all digits are distinct and none is zero, then each digit appears equally often in each position.

Example 7

Find the sum of all three digit numbers formed using the digits $1,2,3$, with each digit used exactly once.

There are

numbers.

Each digit appears in each position

times.

The sum of the digits is

So the contribution to the hundreds place is

The contribution to the tens place is

The contribution to the units place is

So the total sum is

Since

the sum is

Arrangements With Leading Zero

When arranging digits, leading zero is a common issue.

A string like

is not a four digit number.

So if zero is included, handle the first digit separately or subtract invalid arrangements.

Example 8

How many four digit numbers can be formed using the digits $0,1,2,3$, each exactly once?

There are $4!$ arrangements of the four digits.

But those beginning with $0$ are not four digit numbers.

If $0$ is first, the remaining digits can be arranged in

ways.

So the number of valid four digit numbers is

Key Contest Insight

Arrangement problems are about positions.

Rank problems count how many arrangements come before a given arrangement.

Sum problems count how much each digit contributes to each position.

When the objects are repeated, always divide by the factorials of repeated counts.

Strategy Tips

• For distinct objects, start with $n!$
• For repeated objects, divide by the factorials of repeated counts
• For 'together' restrictions, use blocks
• For 'not together' restrictions, consider complement counting
• For rank, count arrangements before the target one position by position
• For sums of arrangements, use positional contribution
• Be careful with leading zero

Common Pitfalls

• Forgetting to divide for repeated letters
• Treating repeated letters as distinct
• Forgetting to add $1$ when computing rank
• Counting rank in the wrong alphabetical order
• Allowing leading zero in a number
• Trying to list all arrangements instead of using positional contribution

Practice Problems