🔀 Permutations and Combinations

1. Fundamental Principle of Counting

If an event can occur in 'm' different ways, following which another event can occur in 'n' different ways, then the total number of occurrences of the events in the given order is m × n.

❗ Factorial Notation (n!)

The continued product of first n natural numbers is called n factorial. n! = n × (n-1) × ... × 3 × 2 × 1.

By definition, 0! = 1.

2. Permutations (Arrangements)

A permutation is an arrangement in a definite order of a number of objects taken some or all at a time.

nPr = n! / (n - r)! Number of permutations of 'n' different things taken 'r' at a time (without repetition).

Permutations with Repetition

The number of permutations of n objects, where p₁ objects are of one kind, p₂ are of second kind, etc., is: n! / (p₁! × p₂! × ...).

Example: Words formed from "MISSISSIPPI".

3. Combinations (Selections)

A combination is a selection of objects where order does NOT matter.

nCr = n! / [r! (n - r)!] Number of combinations of 'n' different things taken 'r' at a time.

Key Properties of nCr

1. nCr = nC(n-r)

2. If nCa = nCb, then either a = b or a + b = n.

3. nCr + nC(r-1) = (n+1)Cr (Pascal's rule).