# Arrangement and selection problems

## Arrangements

• Arrangements involve ordering a number of distinct objects.
• Permutations are an example of arrangements. If we have `n` objects, there are `n!` (factorial) ways to arrange them.
• If we are arranging only `r` objects out of `n` different objects, this is denoted as `_nP_r` and can be calculated using the formula `_nP_r = n!/(n-r)!`
• The factorial function, denoted by `n!`, signifies the product of all positive integers up to `n`. For instance, `5! = 5 × 4 × 3 × 2 × 1 = 120`
• Zero factorial (0!) is unique and is defined as 1.

## Selections

• Selections entail choosing a number of things without regarding the order.
• Selections are often referred to as combinations. Combinations consider the selection of `r` objects from a larger set of `n` objects, without regard to order.
• This is represented as `_nC_r` or `n choose r`, and is calculated using the formula `_nC_r = n!/[r!(n-r)!]`

## Arrangement with Repetition

• When arranging a set of objects where some objects are identical (repeated), we divide the factorial of the total number of objects (`n`) by the factorials of the number of each type of repeat (`r1!, r2!, r3!`, etc.). This is also known as permutations with repetition.
• For instance, in arranging the word `STATISTICS`, we consider the repetitions of “S” (3 times), “T” (3 times), and “I” (2 times).

## Combination with Repetition

• When repetition is allowed while selecting `r` items from `n` types, we calculate combinations using the formula `_n+r-1C_r`.
• This is often found in problems where you are allowed to select the same item more than once.

## Binomial Theorem

• The concepts of arrangements and selections are fundamentally embedded in the Binomial Theorem, where `_nC_r` is key in calculating the coefficients of `(a+b)^n`.
• `_nC_r` in the Binomial Theorem also represents the number of different ways we can choose `r` items from `n` total items.

## The Principle of Inclusion and Exclusion

• The principle of inclusion and exclusion is a counting method to find the number of elements in the union of several sets.
• The principle is based on summing the sizes of the individual sets and then adjusting for overlaps.

Remember that permutations and combinations are key tools in solving arrangement and selection problems. Be sure to thoroughly understand the unique cases these tools apply to as well as their formulas. Practice problems will aid in growing comfortable with incorporating these techniques.