Permutations and Combinations - Permutations

Fundamental Principle of Counting (Multiplication Principle): If one event can occur in $m$ different ways, and following it, another event can occur in $n$ different ways, then the total number of ways both events can occur in the specified order is $m \times n$. Visual: Imagine a tree diagram where the first choice creates $m$ main branches, and each of those branches splits into $n$ smaller sub-branches, resulting in $m \times n$ total paths.

Factorial Notation ($n!$): The product of the first $n$ natural numbers is called $n$ factorial, denoted by $n!$. It is defined as $n! = n \times (n-1) \times (n-2) \times \dots \times 1$. By convention, $0! = 1$. Visual: Think of this as a sequence of numbers decreasing by one at each step, representing the diminishing number of choices available as you fill positions.

Definition of Permutation: A permutation is an arrangement of a specific number of objects in a definite order. Unlike combinations, the order of arrangement is critical. Visual: If you have two colored blocks, Red ($R$) and Blue ($B$), the arrangement $(R, B)$ is visually and mathematically distinct from $(B, R)$ in permutations.

Permutations of $n$ Distinct Objects taken $r$ at a time: When we arrange $r$ objects out of $n$ available distinct objects without replacement, the number of permutations is denoted by $^nP_r$. Visual: Imagine $r$ empty boxes in a row; you have $n$ choices for the first box, $n-1$ for the second, and you continue until all $r$ boxes are filled.

Permutations when Repetition is Allowed: If the objects can be reused (replacement allowed), the number of permutations of $n$ objects taken $r$ at a time is $n^r$. Visual: Imagine a digital lock with 3 dials, where each dial can be any digit from 0-9. Every dial always has 10 options regardless of what the previous dial was set to.

Permutations of Objects Not All Distinct: The number of permutations of $n$ objects, where $p$ objects are of one kind, $q$ objects are of a second kind, and the rest are all different, is given by $\frac{n!}{p!q!}$. Visual: If you arrange the letters in 'BOO', the two 'O's are identical. Swapping them doesn't change the look of the word, so we divide the total arrangements ($3!$) by the ways to arrange the identical items ($2!$) to remove duplicates.

Constraints in Permutations: Sometimes certain items must stay together or certain items must never be together. Visual: If items A and B must be together, we 'tie' them into a single block. This block and the remaining items are then arranged like a single unit, and then A and B are arranged within their own internal block.