Permutation and Combinations

Permutation and combination are basically methods that are used to depict a distinct group of objects, by selecting them from a group and then creating subsets with them. It is a way to define the several ways that are used to arrange a specific group of data. When the person selects the objects or the data from that specific group, the process is known as permutations, and the order through which these objects are represented is known as combinations. There are numerous permutation and combination formulas that can be used along with the concepts, which we have discussed below. 

Both concepts are very important aspects of physics as well as mathematics. Here we will be discussing the concepts of permutation and combination thoroughly, stating the difference between the two, and illustrating examples and problems based on the concepts as well.

The Concepts Of Permutation and Combination

When it comes to physics, the concept of permutation refers to the act where one organises the constituents of a specific set to resemble an order, or a sequence. If said in other words, in the case that the set is already in an ordered manner, then rearranging its components to resemble a specific order is known as permutation. Cases of permutations occur in almost every area of physics or mathematics, in some way or the other. They usually arise when specific numbers come in some particular finite sets.

Next is the concept of combinations, which is where certain items are selected from a set or collection, in a way that the order in which the selection occurs does not matter. There lies the difference of combination with permutation. If it is a small set, then it is even possible to count these numbers. The combination can also be defined as the combination of a n number of objects that have been selected k at a time but does not have repetition. However, in the case of a combination where repetition has been allowed to happen, the term “k” combination or “k” selection can also be used.

The Formulas Of Permutation and Combination

There are several formulas that can be used along with the concepts of permutation and combination. The permutation and combination formulas meanings have been discussed below. There are two main formulas:

The Formula of Permutation

Permutation refers to the selection of r objects that have been selected from a set of n objects which has no repetition and where the order must matter. 

n P r = ( n ! ) / (n – r ) !

The Formula of Combination 

Combination is when the choice of “r” objects from a group of “n” objects has no repetition and are in a case that the order does not matter. 

n C r = (n r) = nP r / r ! = n ! / r ! (n – r) !

The Difference Between Permutation and Combination

Solved Problems Based on the Permutation and Combination Formulas

We have discussed a few permutations and combinations formulas questions below:

Problem 1

In the case that n is 12 and r is 2, find the total number of permutations and combinations that can happen. 

Solution

If it is given that n is 12 and r is 2, and if we use the formula that we have mentioned above:

Permutation formula –nP r = ( n ! ) / ( n – r ) ! = ( 12 ! ) / ( 12 – 2 ) ! = 12 ! / 10 ! = ( 12 x 11 x 10 ! ) / 10 ! = 132; 

Combination formula – nC r = n 1 / r ! ( n – r ) ! =12 ! / 2 ! ( 12 – 2 ) ! = 12 ! / 2 ! ( 10 ) ! = 12 x 11 x 10 ! / 2 ! ( 10 ) ! = 66.

Problem 2

You take a dictionary and find that all the permutations of the alphabets of the word “again” have been arranged in a specific order. Find out what the 49th word in it can be.

Solution

If we start with the letter “a” and arrange the other four alphabets, that is g, a , i , n, then it will be 4 !, that is 24. It comes to the first 24 words.

If we start with the letter g and arrange the other four alphabets, that is a, a , i , n, then it is 4 ! / 2 ! That is 12. It comes to the next 12 words.

If we start with the letter i and arrange the other four alphabets, that is a, a , g , n, then it is 4 ! / 2 ! That is 12. It comes to the next 12 words.

This overall procedure will come up to the word 48. Thus the word “naagi” will be the 49th word.

Conclusion

This article explains permutation and combination formulas with examples. When the person selects the objects or the data from that specific group, the process is known as permutations, and the order through which these objects are represented is known as combinations. The permutation and combinations formulas are nPr = n ! / (n – r) ! And nCr = n ! / [ r ! ( n – r ) ! ] respectively.