Know about the Combinations in Mathematics

An Overview: Combinations 

The grouping or sub-grouping of elements or items for a given set of numbers or items or elements is called a Combination. 

In simple words, a combination of a set of distinct objects is defined as an unordered selection of elements from the set.

When performing solutions related to Combinations, the number of ways of the selection of items in which order the elements or items of a given group are selected does not matter.

Combinations Formula

n = Number of items/elements in a given group or set.

r = Number of items selected from the given set/group.

C (n,r) = nCr = n!r! n-r!

About Permutations

The arrangement of items or elements of a given group in a mathematical problem, in a specific systematic order, is called Permutations. It is the order in which the number of elements or items of a given group are selected and arranged matters. 

The formula for Permutation

n = Number of items/elements in a given group or set.

r = Number of items selected from the given set/group.

P (n,r) = nPr = n! n-r!

Types of Combinations in Mathematics

There are two types of Combinations in Mathematics (One must keep in mind the order of the selected items/elements from the set does not matter).

Combinations in which Repetition is Allowed- The coins in an individual’s pocket are (1, 1, 1, 2, 2, 5, 5, 5, 10, 10, 10, 10, 10, 20, 20, 20, 20).

Combinations in which Repetition is Not Allowed- The order numbers in McDonalds are (14, 20, 21, 24, 27, 28, 29, 35, 29, 44, 45, 49, 50, 51, 59, 66).

Combinations in which Repetition is Allowed

The number of items selected from a set can be the same, or they can repeat in a combination where repetition is allowed.

Number of Combinations of ‘r’ objects chosen from ‘n’ objects- repetition is allowed.

The formula used is – n + r – 1 !r! n – 1 !

Combination in which Repetition is Not Allowed

The number of items selected from a set cannot be the same, or they cannot repeat in a combination where repetition is not allowed.

Number of Combinations of ‘r’ objects chosen from ‘n’ objects- repetition is not allowed.

The formula used is – n !r! n – r !

Types of Permutations in Mathematics

1. The ‘n’ number of objects are different from each other, and repetition is allowed in this part or ‘r’ objects which are chosen from the set or group. 

For Example – An individual rolls a die five times, each time he rolls the die he will only get a number from 1 to 6 because the maximum space of the die is from 1 to 6. There is no number above 6. Thus, the numbers can be repeated every time the die is rolled.

When repetition is allowed- ‘P’ is a permutation or arrangement of ‘r’ elements/items from a given group of elements or set of ‘n’ things with repetition allowed.

The formula used is nPr = nr

1. The ‘n’ number of objects are different from each other, and repetition is not allowed in this part or ‘r’ objects which are chosen from the set or group, r < n or r = n.

For Example – Cards from a deck of cards are picked up without keeping them in the deck. In this way, the outcome will never repeat as the same cards will not be picked.

When repetition is not allowed- ‘P’ is a permutation or arrangement of ‘r’ elements/items from a given group of elements or a set of ‘n’ things without replacement.

The formula used is: nPr =  n! n-r !

1. The ‘n’ objects are not different from each other, all the items are used in the arrangement. 

Difference between Permutation and Combination

Conclusion

Permutations and Combinations are simple to understand if learnt with interest. If an individual is interested in Probability as a Mathematical subject, then he/she will surely be interested and grasp this topic easily.

Permutations and combinations go hand in hand so one must know all the formulae and go on with the given problems.