nPr Formula — Permutations, Derivation, Examples

What Is the nPr Formula?

A permutation is an ordered selection. The nPr formula gives the number of ways to pick and arrange $r$ items from $n$ distinct items, where rearranging the chosen items counts as a new result.

$$\boxed{;^nP_r = \frac{n!}{(n-r)!};}$$

The condition that the $n$ objects are distinct matters — the formula assumes no repeats among the items, and that an item, once placed, is not used again. Permutations are the ordered half of combinatorics; the unordered half is counted by nCr, the combination formula.

How Is the nPr Formula Derived?

The derivation comes straight from the fundamental counting principle: if one choice has $x$ options and the next has $y$, the two together have $x \times y$ outcomes. Filling $r$ ordered positions from $n$ objects works the same way.

• The first position can be filled $n$ ways.

• The second position then has $n - 1$ objects left.

• The third has $n - 2$, and so on.

• The $r$-th (last) position has $n - r + 1$ objects left.

Multiply the choices:

$$^nP_r = n(n-1)(n-2)\cdots(n-r+1).$$

That product is the top of a factorial cut short after $r$ terms. To write it cleanly, multiply and divide by $(n-r)!$:

$$^nP_r = \frac{n(n-1)\cdots(n-r+1)\cdot(n-r)!}{(n-r)!} = \frac{n!}{(n-r)!}.$$

The $(n-r)!$ in the denominator is exactly the part of $n!$ you do not use — the objects you never placed. That is the whole formula, and it is why $^nP_n = \frac{n!}{0!} = n!$: arranging all $n$ objects uses every factor.

Examples of the nPr Formula

Example 1

Evaluate $^{10}P_4$.

$$^{10}P_4 = \frac{10!}{(10-4)!} = \frac{10!}{6!} = 10 \times 9 \times 8 \times 7 = 5040.$$

Final answer: $5040$.

Example 2

How many 3-letter arrangements can be made from the letters of the word MATH (no repeats)?

Wrong attempt. A student thinks "choose 3 letters from 4" and computes $\frac{4!}{3!,1!} = 4$ using the combination formula. Check it against a small case: from M, A, T the arrangements MAT and TAM are clearly different words, yet the combination count treats them as one. Order matters here, so a combination undercounts.

Correct. Use the permutation formula:

$$^4P_3 = \frac{4!}{(4-3)!} = \frac{4!}{1!} = 24.$$

Final answer: $24$ arrangements.

Example 3

In how many ways can 8 runners finish 1st, 2nd, and 3rd in a race?

Three ordered positions from eight runners:

$$^8P_3 = \frac{8!}{5!} = 8 \times 7 \times 6 = 336.$$

Final answer: $336$ ways.

Example 4

How many ways can 6 books be arranged on a shelf?

All six are arranged, so $r = n = 6$:

$$^6P_6 = \frac{6!}{0!} = 720.$$

Final answer: $720$.

Example 5

How many 4-digit numbers can be formed from the digits 1–9 with no digit repeated?

Nine distinct digits, four ordered positions:

$$^9P_4 = \frac{9!}{5!} = 9 \times 8 \times 7 \times 6 = 3024.$$

Final answer: $3024$.

Example 6

Five people sit in a row of five seats, but two specific friends must sit at the two ends. How many arrangements are possible?

Fix the two friends at the ends: they can swap, giving $^2P_2 = 2$ ways. The remaining three people fill the three middle seats in $^3P_3 = 6$ ways. By the counting principle, total $= 2 \times 6 = 12$.

Final answer: $12$ arrangements.

Why Permutations Matter — From Passwords to PIN Pads

Counting ordered arrangements was forced into existence by problems where sequence is the answer, not a detail.

• Security. The number of possible PINs, passwords, and lock combinations is a permutation count — a 4-digit PIN with no repeats has $^{10}P_4 = 5040$ possibilities, which is exactly why short codes are weak.

• Scheduling and rankings. Seating charts, tournament brackets, race results, playlist orders — anything where "who is first" changes the outcome — is permutation territory.

• Genetics and biology. The order of bases along a DNA strand is a permutation problem; sequence determines the gene.

• Cryptography. Modern ciphers rest on how astronomically many ways symbols can be ordered, so that guessing the right order is infeasible.

The destination, in a later probability chapter, is computing the odds of a specific arrangement: once you can count the arrangements, the probability of any one of them is one divided by that count. Counting cleanly is also what makes other reference formulas tractable — the same factorial machinery sits inside the sum of arithmetic sequence formula when you count terms before adding them.

Tripping Points to Avoid

Mistake 1: Using nPr when order does not matter

Where it slips in: Problems about committees, teams, handshakes, or lottery picks — where rearranging the chosen people changes nothing.

Don't do this: Apply $^nP_r$ to "choose 3 students for a committee from 10." A committee of {A, B, C} is the same committee as {C, B, A}.

The correct way: When order is irrelevant, use the combination formula $^nC_r = \frac{n!}{r!,(n-r)!}$.

Mistake 2: Mishandling $0!$

Where it slips in: Cases where $r = n$, so the denominator is $0!$.

Don't do this: Treat $0!$ as $0$, which would make $^nP_n$ undefined.

The correct way: $0! = 1$ by definition. So $^nP_n = \frac{n!}{0!} = n!$, which matches the count of arranging all $n$ objects.

Mistake 3: Forgetting the "distinct objects, no repetition" assumption

Where it slips in: Words with repeated letters (like BALLOON) or problems allowing repeats.

Don't do this: Apply plain $^nP_r$ to arrange the letters of a word with duplicates.

The correct way: When items repeat, divide by the factorial of each repeat count, or use the with-repetition rule $n^r$ when repeats are allowed. The bare nPr formula assumes all $n$ items are different and none is reused.

Mistake 4: Subtracting in the wrong place

Where it slips in: Plugging into $\frac{n!}{(n-r)!}$ in a hurry.

Don't do this: Compute $\frac{(n-r)!}{n!}$ or $\frac{n!}{r!}$ by misremembering which factorial sits where.

The correct way: The bigger factorial $n!$ is on top; the denominator is $(n-r)!$.

Conclusion

• The nPr formula counts ordered arrangements of $r$ from $n$ distinct objects: $^nP_r = \frac{n!}{(n-r)!}$.

• It is derived from the counting principle: $n$ choices for the first place, $n-1$ for the second, down to $n-r+1$ for the last.

• Use nPr when order matters and nCr when it does not — swapping two chosen items being a new result is the test.

• The most common mistake is using nPr for an unordered selection like a committee.

• Permutations underpin passwords, schedules, rankings, and the probability of a specific arrangement.

Practice These Three Before Moving On

1. Evaluate $^7P_2$.

2. How many ways can 5 students be seated in 3 chairs?

3. How many 5-letter arrangements use the distinct letters of PRIME?

Answer to Question 1: $42$. Solve Questions 2 and 3 with $\frac{n!}{(n-r)!}$; if you are unsure whether order matters, return to Mistake 1.

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