Question

What is ${}^n{C_r}$ in probability?

Answer 1

${}^n{C_r}$ represents the collection of objects from a category of objects that do not import the order of objects. The order of objects does matter in case of combination. ${}^n{C_r}$ is calculated with the help of following formula:
${}^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}$

Complete step by step solution:
In probability, we use permutation and combination to solve the problem easily. We can term permutation and combination as two functions that are used to solve probability problems easily and also in less time.
We are asked about the combination ${}^n{C_r}$ in probability in the given problem, so we will discuss only about combinations, not permutation.
Combination is the various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a combination when order is not a factor. Combination of $r$ items from a total of $n$ items is represented as ${}^n{C_r}$
And also ${}^n{C_r}$ is calculated as ${}^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}$
We use combinations in probability when we have to find the number of ways to choose a sample of $r$ elements from a set of $n$ distinct objects where order does not matter. It is used in both cases when repetition is allowed and when it is not allowed.
Let us understand this with an example,
What is possibility of choosing $4$ students in class of $30$
This will be calculated as ${}^{30}{C_4} = \dfrac{{30!}}{{(30 - 4)!4!}} = 27405$

Note : If you don’t care what order you have things, it’s a combination. Think of combining ingredients, or musical chords. Lottery tickets, where you pick a few numbers, are a combination. That’s because the order doesn’t matter (but the numbers you select do).