Permutations and Combinations Concepts

Understanding the Concepts of Permutations and Combinations

You can now understand the fundamental concepts of Permutations and Combinations:

In mathematics:

• When the sequence is irrelevant, it is a Combination.
• When the order is important, it is a Permutation.

1. Permutations

A permutation is a defined order arrangement of a variety of items taken one or all at once.

Consider the following ten numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. P(10,4) = 5040,

The number of different 4-digit-PIN which can be formed using these 10 numbers is 5040. P(10,4) = 5040

The permutations of 4 numbers taken from 10 numbers equal to the factorial of 10 divided by the factorial of the difference of 10 and 4.

Permutation Formula

If the total quantity of data is 'n' and the option is 'r,' then permutation will be (without replacement and without concern for order)

nPr = (n!) / (n-r)!

2. Combinations

We use the word 'combination' loosely in English, without considering whether the sequence of items is relevant.

Example 1

'My fruit salad consists of apples, grapes, and bananas,' for example. It doesn't matter what order the fruits are in; it could be 'bananas, grapes, and apples' or 'grapes, apples, and bananas'; it's the same fruit salad.

Example 2

'The combination to the safe is 472,' for example. We are now concerned about the order. '724' and '247' will not work. It has to be 4-7-2 precisely.

Combination Formula

The selection of 'r' items from a group of 'n' data without consideration for order or replacement-

nCr = nPr / r! = n! / {r! (n-r)!}

Listed below are the different Permutations and Combinations concepts:

The fundamental principle of counting** is the underlying concept behind permutations and combinations, which states that if there are multiple options, multiply the supplied terms and add those numerous terms if there is a single selection.

In layman's terms, if we see or understand the word 'AND' in the problem, just 'MULTIPLY' or if we see the word 'OR' in the problem, simply 'ADD' everything.

For Example: There are eight different shirts and five different pants, how many differing ways can a person dress?

We can answer this problem in a variety of ways, including visualising all conceivable combinations such as shirt 1 pant 1, shirt 1 pant 2, and so on.

However, there is a shortcut method: a person must choose a shirt and a pair of pants to dress up.

Here AND is the crucial word. So just multiply the given numbers.

8 * 5 = 40 is the total number of ways.

For Example: If there are eight different bikes and five different vehicles, how many different ways can someone use the vehicles to get to their destination?

In the preceding example, we can solve this problem in a variety of ways, including visualising all possible combinations.

However, there is a shortcut method: a person must choose between a bike and a car.

Here OR is the key word. So simply add the given numbers

8 + 5 = 13 is the total number of ways.