Permutation and Combination Calculator

Understanding Permutations and Combinations

What are Permutations and Combinations?

Permutations and combinations are fundamental concepts in probability theory and statistics. They help us calculate the number of ways to select or arrange items from a larger set.

Permutations (nPr)

A permutation is an arrangement of objects where order matters. The formula for permutations is:

\[P(n,r) = \frac{n!}{(n-r)!}\]

Where:

• n = total number of items
• r = number of items being arranged
• ! denotes factorial (e.g., 5! = 5 × 4 × 3 × 2 × 1)

Combinations (nCr)

A combination is a selection of objects where order doesn't matter. The formula for combinations is:

\[C(n,r) = \frac{n!}{r!(n-r)!}\]

Where:

• n = total number of items
• r = number of items being selected
• ! denotes factorial

Example Calculations

Permutation Example:

Calculate 5P3 (permutation of 5 items taken 3 at a time)

\[P(5,3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{5 \times 4 \times 3 \times 2!}{2!} = 5 \times 4 \times 3 = 60\]

Combination Example:

Calculate 5C3 (combination of 5 items taken 3 at a time)

\[C(5,3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{5 \times 4 \times 3!}{3! \times 2 \times 1} = 10\]

Visual Representation

This diagram illustrates the key differences between permutations and combinations, using the example of selecting 3 items from a set of 5.