Permutation & Combination Calculator — nPr and nCr

Total Items (n)

020

Items Chosen (r)

020

Permutations (nPr) — order matters

10P3 = 10! / (10−3)!

Combinations (nCr) — order doesn't matter

10C3 = 10! / (3! × (10−3)!)

0

Factorials Used

10!1

3!1

(10−3)! = 7!1

What is a nPr nCr?

The Permutation and Combination Calculator computes both nPr (permutations — ordered arrangements) and nCr (combinations — unordered selections) simultaneously for any values of n (total items) and r (items chosen). It also displays the key factorials n!, r!, and (n−r)! used in the formulae, along with a full step-by-step calculation breakdown.

Permutations and combinations are the mathematical tools for counting — answering questions like "how many ways can 10 runners be assigned medals?" (permutation: order matters) or "how many 4-person committees can be formed from 12 candidates?" (combination: order doesn't matter). The fundamental distinction is whether the sequence or arrangement of the chosen items matters to the problem.

In the context of probability, knowing how to count favourable and total outcomes using combinations is essential. The number of ways to draw 5 cards from a deck (52C5 = 2,598,960) is the total-outcomes denominator for any 5-card hand probability. Use this calculator alongside the Probability Calculator to first count outcomes, then compute the probability.

These topics appear in the CBSE Class 11 Mathematics syllabus (Chapter 7: Permutations and Combinations) and are a mandatory part of the JEE Main and JEE Advanced Mathematics section.

How to use this nPr nCr calculator

Set Total Items (n) — use the slider or type the total number of distinct items available. For a class of 30 students, n = 30. Range is 0 to 20.
Set Items Chosen (r) — use the slider or type how many items are being selected or arranged. Ensure r ≤ n; the slider automatically caps r at the current n value.
Read Permutations (nPr) — the dark result card shows nPr with the formula substitution. Use this if order matters in your problem.
Read Combinations (nCr) — the white result card below shows nCr. Use this if order does not matter (committees, teams, selections).
Check the Factorials panel — verify n!, r!, and (n−r)! individually to confirm you are applying the correct formula to your problem.

Formula & Methodology

Permutation:nPr = n! / (n − r)!

Combination:nCr = n! / (r! × (n − r)!)

Relationship:nPr = r! × nCr

Key factorial values:
- 0! = 1 (by definition)
- 1! = 1
- 5! = 120
- 10! = 3,628,800
- 15! = 1,307,674,368,000
- 20! = 2,432,902,008,176,640,000

Variable definitions:
- n — total number of distinct items
- r — number of items to select or arrange (0 ≤ r ≤ n)
- n! — n factorial = product of all integers from 1 to n
- (n−r)! — factorial of remaining (unchosen) items

Worked example — forming a cricket team:

A cricket club has 18 players. They need to select 11 players for a match (combination) and then assign the captaincy and vice-captaincy (permutation within the 11).

Step 1 — Combinations (selecting 11 from 18, order irrelevant):18C11 = 18! / (11! × 7!) = 31,824 ways to select the team

Step 2 — Permutations within selected 11 (assigning captain and vice-captain):11P2 = 11! / (11−2)! = 11 × 10 = 110 ways to assign the two roles

Step 3 — Total arrangements (team selection × role assignment):31,824 × 110 = 3,500,640 distinct captain-and-vice-captain configurations

This two-stage counting problem illustrates the multiplication principle: first combine, then permute within the selection.

Assumption: All n items are distinct (no two identical). Permutation and combination formulas change when items are repeated or non-distinct — those cases (circular permutations, multinomial coefficients) are not covered by the standard nPr and nCr formulas.

Frequently Asked Questions

What is the difference between permutation and combination?

Permutation counts arrangements where order matters — selecting 3 people for President, Secretary, and Treasurer from 10 candidates gives a different result depending on who gets which role. Combination counts selections where order does not matter — choosing any 3 people from 10 for a committee, regardless of internal ordering. The same set of r items produces r! times more permutations than combinations.

What is the formula for nPr (permutation)?

The permutation formula is nPr = n! / (n−r)!, where n is the total number of items and r is the number being arranged. It counts the number of ways to arrange r items chosen from n distinct items where order matters. For n=5 and r=2: 5P2 = 5!/(5−2)! = 120/6 = 20 arrangements.

What is the formula for nCr (combination)?

The combination formula is nCr = n! / (r! × (n−r)!), where n is the total items and r is the number being selected. It counts selections where order does not matter. For n=5 and r=2: 5C2 = 5!/(2!×3!) = 120/(2×6) = 10 selections. nCr is always a whole number and always less than or equal to nPr for the same n and r.

How does the Permutation and Combination Calculator work?

Enter n (total items) and r (items chosen) using the sliders. The calculator instantly computes both nPr (permutations) and nCr (combinations), along with the factorials n!, r!, and (n−r)! for reference. The step-by-step breakdown shows the full formula substitution. Both results appear side by side to highlight the difference between ordered and unordered selection.

Why is nCr always less than or equal to nPr?

nPr = nCr × r!, so nPr is nCr multiplied by the number of ways to arrange the r selected items among themselves. When r=1, nPr = nCr (only one way to arrange one item). When r=2, nPr = 2 × nCr. When r=3, nPr = 6 × nCr. This relationship explains why 'choosing without order' always yields fewer or equal results than 'arranging with order'.

What happens when r equals 0 or r equals n?

When r=0, both nP0 = 1 (there is exactly one way to arrange zero items — do nothing) and nC0 = 1 (exactly one way to choose nothing). When r=n, nPn = n! (the total number of ways to arrange all n items) and nCn = 1 (only one way to select all items). These are the boundary cases that confirm the formulas work consistently across all valid values of r.

What is a factorial and why does it appear in permutation and combination formulas?

A factorial, written n!, is the product of all positive integers from 1 to n: n! = 1 × 2 × 3 × ... × n. It counts the number of ways to arrange n distinct objects in a sequence. Permutations and combinations use factorials to count arrangements efficiently — nPr divides n! by the arrangements of unchosen items, and nCr additionally divides by the internal arrangements of chosen items.

Are permutations and combinations part of the JEE syllabus?

Yes — Permutations and Combinations is a dedicated chapter in the JEE Main and JEE Advanced Mathematics syllabus, and also appears in CBSE Class 11 (Chapter 7). JEE problems often involve multi-step counting arguments requiring both nPr and nCr, along with circular arrangements, repetition, and inclusion-exclusion. Knowing the relationship nPr = r! × nCr is frequently tested.

How do I use combinations to find probability?

Probability of an event = (favourable combinations) / (total combinations). For example, the probability of getting exactly 2 heads in 5 coin tosses is 5C2 / 2⁵ = 10/32 = 5/16. First compute nCr using this calculator to count favourable and total outcomes, then divide using the [Probability Calculator](/probability-calculator/) or manually. This is the classical combinatorial approach to probability.

What is the maximum n value this calculator supports?

The calculator supports n up to 20, which keeps factorials within the range of standard number representation (20! ≈ 2.43 × 10¹⁸). For n above 20, factorials exceed the precision of standard floating-point arithmetic and results become approximate. For very large n with small r, the combination formula can be computed more accurately using logarithms or the multiplicative formula nCr = n×(n−1)×...×(n−r+1) / r!, which avoids computing large factorials.

What is the difference between sampling with and without replacement?

Sampling without replacement means each item can be chosen at most once — this is the standard permutation/combination scenario. Sampling with replacement means an item can be chosen multiple times, so there are nʳ equally likely ordered outcomes (instead of nPr). For example, rolling a die twice has 6² = 36 outcomes with replacement, but only 6P2 = 30 ordered outcomes without replacement.