Simple Factorial (n!)

The factorial of n is the product of all positive integers less than or equal to n.

Formula: n! = n × (n-1) × (n-2) × ... × 2 × 1

Example: 5! = 5 × 4 × 3 × 2 × 1 = 120

Special case: 0! = 1 by definition

Combinations (nCr)

The number of ways to choose r objects from n objects when order doesn't matter.

Formula: C(n,r) = n! / (r! × (n-r)!)

Example: How many ways can you choose 3 cards from 5 cards? C(5,3) = 10

Uses: Lottery combinations, team selection, card hand probabilities

Permutations (nPr)

The number of ways to arrange r objects from n objects when order matters.

Formula: P(n,r) = n! / (n-r)!

Example: How many ways can you arrange 3 books from 5 books? P(5,3) = 60

Uses: Race finishing positions, password combinations, seating arrangements

Factorial Ratio (n! / m!)

Calculates the ratio of two factorials, useful in various mathematical formulas.

Formula: n! / m! = n × (n-1) × ... × (m+1) when n > m

Example: 10! / 5! = 10 × 9 × 8 × 7 × 6 = 30,240

Important Notes

• Factorials grow extremely quickly - 20! is already over 2 quintillion
• Maximum input is typically 170 due to computational limits
• For combinations and permutations, r must be ≤ n
• Results may be displayed in scientific notation for very large numbers

Common Applications

• Probability and statistics calculations
• Lottery and gambling odds
• Combinatorial optimization problems
• Binomial theorem expansions
• Graph theory and network analysis
• Cryptography and code breaking