Permutation and Combination Calculator

What are Permutations and Combinations?

Permutations and combinations are fundamental concepts in probability and statistics, dealing with the arrangement and selection of items from a set.

Formulas and Their Meanings

1. Permutation (\(P(n,r)\)): \[P(n,r) = \frac{n!}{(n-r)!}\] Where \(n\) is the total number of items and \(r\) is the number of items being arranged.

2. Combination (\(C(n,r)\)): \[C(n,r) = \frac{n!}{r!(n-r)!}\] Where \(n\) is the total number of items and \(r\) is the number of items being chosen.

Calculation Steps

1. Identify the values of n (total items) and r (items chosen/arranged).
2. For Permutation: Calculate \(n!\) and \((n-r)!\), then divide.
3. For Combination: Calculate \(n!\), \(r!\), and \((n-r)!\), then divide as per the formula.

Example Calculation

Let's calculate for n = 5 and r = 3

1. Permutation: \(P(5,3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = 60\)
2. Combination: \(C(5,3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = 10\)

Visual Representation

This diagram illustrates the example where we have 5 total items (outer circle) and we're choosing 3 items (inner circle). The permutation (60) represents the number of ways to arrange 3 items from 5, while the combination (10) represents the number of ways to choose 3 items from 5 without regard to order.