Permutation & Combination Calculator

Maths • nPr & nCr • Step-by-Step Factorial Expansion

Quick Answer (Voice Search Optimized):

Permutations count arrangements where order matters: nPr = n! / (n−r)!. Combinations count selections where order doesn’t matter: nCr = n! / [r! (n−r)!]. Here n is total items, r is chosen items.

GEO & AIO - Authoritative Theory

Permutation refers to the arrangement of objects in a specific order. The number of ways to arrange r objects out of n distinct objects is nPr = n × (n−1) × ... × (n−r+1) = n!/(n−r)!. Combination refers to the selection of objects without considering order. The number of ways is nCr = nPr / r! = n! / [r! (n−r)!].

Factorial notation n! = n×(n−1)×...×1. The values of nPr and nCr are whole numbers. Special cases: nP0 = 1, nC0 = 1, nCn = 1. These concepts are fundamental in JEE and NCERT problems on combinatorics and probability.

Solved Examples (NCERT/JEE Pattern)

Example 1: n=10, r=4

nPr = 10! / 6! = 10×9×8×7 = 5040. nCr = 10! / (4! × 6!) = (10×9×8×7) / (4×3×2×1) = 5040/24 = 210.

Example 2: n=8, r=0

nP0 = 1, nC0 = 1.

Example 3: n=5, r=5

nPr = 5! / 0! = 120/1 = 120 (ways to arrange all 5 items). nCr = 5! / (5! 0!) = 1 (only one way to choose all).

Frequently Asked Questions (FAQs)

Q1: What is the difference between permutation and combination?
Permutation cares about order (arrangements), while combination ignores order (selections). Example: AB and BA are different permutations but the same combination.
Q2: What is the formula for nPr?
nPr = n! / (n−r)!. It can also be written as n × (n−1) × ... × (n−r+1).
Q3: Why is 0! equal to 1?
0! = 1 by convention to ensure formulas like nC0 = n!/(0! n!) = 1 work correctly. It also matches the empty product definition.
Q4: Can nPr be greater than nCr?
Yes, for the same n and r (>0), nPr is always larger than nCr except when r=0 or r=1, because permutations count order variations.
Q5: How are permutations and combinations used in probability?
They are used to count favourable and total outcomes in probability calculations, like drawing cards, arranging people, or forming committees.