Problem 78 - Algebra and Trigonometry

Question

What is a combination?

Step-by-Step Solution

Answer

A combination, in probability and statistics, is a selection of all or part of a collection of things, in which the order doesn't matter. It can be calculated using the formula: \( C(n, r) = \frac{n!}{r!(n-r)!} \).

Step 1: Defining Combination

A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected.

Step 2: Illustrating with an Example

For instance, if you have 5 friends and you want to choose 3 of them to go to a concert, you would use combinations to find how many different ways you can choose them. It does not matter what order you choose your friends because you all will meet at the concert at the same time.

Step 3: Mathematical Representation of Combination

Mathematically, combination is represented as \(C(n, r)\) or \(_nC_r\) or \((^{n}_{r})\) where \(n\) is the total number of items to choose from, and \(r\) is the number of items to choose. It's calculated using the formula: \[C(n, r) = \frac{n!}{r!(n-r)!}\] Here \(n!\) denotes the factorial of \(n\), which is the product of all positive integers less than or equal to \(n\).