Permutation Example- Counting Arrangements in Probability

What Is a Permutation? The Short Answer

A permutation is an ordered arrangement of items where the sequence matters. If you rearrange anything and the order changes the result, that's a permutation.

Flip a coin three times: HHH, HHT, HTH, THH, THT, TTH, HTT, TTT. Each one is a different permutation. Getting heads then heads then heads is not the same as heads then heads then tails.

That's the core idea. Everything else is just math to count them faster.

The Permutation Formula

When you need to arrange r items from a set of n items, use:

P(n,r) = n! / (n - r)!

The ! symbol means factorial. So 5! = 5 × 4 × 3 × 2 × 1 = 120.

Breaking Down the Formula

For P(5,3):

• 5! = 120
• (5-3)! = 2! = 2
• 120 ÷ 2 = 60 permutations

Or written out: 5 × 4 × 3 = 60. You multiply the first r numbers in the factorial.

Permutation Examples You Can Actually Use

Example 1: Selecting Race Winners

Ten runners compete. How many ways can gold, silver, and bronze be awarded?

P(10,3) = 10! / (10-3)! = 10! / 7! = 10 × 9 × 8 = 720 ways

You don't care about 4th through 10th place. Only the top three spots matter.

Example 2: Password Combinations

A PIN code uses 4 digits from 0-9 without repetition. How many possible PINs?

P(10,4) = 10! / 6! = 10 × 9 × 8 × 7 = 5,040 possible PINs

If repetition was allowed, you'd have 10⁴ = 10,000. Permutations without repetition give fewer options.

Example 3: Arranging Books on a Shelf

You have 6 different books but only room for 4 on a shelf. How many arrangements?

P(6,4) = 6! / 2! = 6 × 5 × 4 × 3 = 360 arrangements

Book A in position 1 with Book B in position 2 is different from Book B in position 1 with Book A in position 2. Order matters.

Permutations vs Combinations: The Difference

People mix these up constantly. Here's the deal:

• Permutation = order matters (ABC ≠ BCA)
• Combination = order doesn't matter (ABC = BCA)

Choosing 3 people to be president, VP, and secretary is a permutation problem. The positions are different.

Choosing 3 people to form a committee is a combination problem. No one has a specific role.

Quick Comparison Table

How to Calculate Permutations: Step by Step

Here's how to work through any permutation problem:

Step 1: Identify n and r

Find the total number of items (n) and how many you're arranging (r).

Step 2: Check for Restrictions

Can items repeat? Most permutation problems in probability assume no repetition. If repetition is allowed, you use n^r instead.

Step 3: Apply the Formula

For P(n,r) with no repetition:

• Write out n! ÷ (n-r)!
• Cancel common terms
• Multiply the remaining numbers

Step 4: Verify Your Answer

For small numbers, list a few arrangements to check if your answer seems reasonable.

Permutations With Repetition

Sometimes items repeat. The word "STATISTICS" has repeated letters.

When all n items are arranged but some are identical, use:

P(n!) / (n₁! × n₂! × ...)

How many ways to arrange the letters in "BANANA"?

• n = 6 (six letters total)
• n₁ = 3 (A appears 3 times)
• n₂ = 2 (N appears 2 times)
• n₃ = 1 (B appears 1 time)

6! / (3! × 2! × 1!) = 720 / 12 = 60 arrangements

When Permutations Show Up in Real Life

• Sports tournaments — seeding brackets, match scheduling
• Cryptography — calculating key spaces for security algorithms
• Lottery analysis — probability of specific number sequences
• Route optimization — delivery order, travel planning
• Quality control — testing components in different sequences

Common Permutation Mistakes

Using combinations when order matters. Always ask: does sequence affect the outcome?

Forgetting to divide by factorials when items repeat. The formula changes.

Miscalculating n and r. Read the problem carefully. "How many ways to pick 3 winners from 100 entries" means n=100, r=3.

Confusing P(n,r) with n!. Use n! only when arranging all items from a set of n.

The Formula to Memorize

P(n,r) = n! / (n - r)!

That's it. Plug in your numbers, cancel what you can, multiply the rest. Every permutation problem uses this structure.

If the problem involves repeated items, divide by the factorial of each repeated item count.