Combinations and their applications

A collection of items can be represented in two ways: permutation and combination. Both are distinct, and many students are perplexed by the distinction. We term it a combination when the order of the elements doesn’t matter. We have a permutation if the order does matter. A permutation is also known as an ordered combination.

The term “combinations” refers to the process of choosing items from a list of possibilities. We have no plans to rearrange things here. We intend to select them. We count the number of unique r-selections among a set of n objects nCr.

Combinations are used when only the number of possible groupings needs to be specified and the order of arrangements isn’t necessary. 

Combination 

The combination is a method of selecting or choosing a different person or object from a group of objects by a person taking one or more than one at a time. Let r be the number of different things, and n be the total number of objects. nCr = nỊ / (n-r)ỊrỊ

nCr is a combination of ‘r’ objects out of total ‘n’ objects.

Formula of combination 

The combinations formula, as well as factorials and permutations, are used to calculate combinations. Consider the following scenario: we have n items at our disposal and we want to know how many different ways we can choose r items from these n items. The total number of permutations of these n things taken r at a time is first calculated. That value would be n P r. Each combination in this list of n P r permutations will be enumerated r! times because r items can be permuted among themselves in r! ways. As a result, n C r refers to the total number of permutations and combinations of these n elements, taken r at a time.

Combination: 1. nCr = n! / (n-r)! r! (with repetition)

2.(r + n − 1)!/r!(n − 1)! (without repetition)

Derivation of formula 

Let us take (r) number of boxes and each box can hold one thing.

Then, the number of possibilities for selecting the first object from a set of n objects= n

The number of possibilities to choose the second object from among (n-1) distinct objects is as follows= (n-1)

Number of possibilities to choose the third object from (n-2) different options: (n-2)

Number of possible ways to choose the rth object from a set of [n-(r-1)] different objects= [n-(r-1)]

An ordered subset of r elements is created by completing the selection of r things from the initial set of n things.

∴ The number of ways to choose r elements from a set of n elements is n (n – 1) (n – 2) (n-3)… (n – (r – 1) or n (n – 1) (n – 2)… (n – r + 1) n – r + 1 n – r + 1

Let’s take a look at the ordered subset of r elements, as well as all of their permutations. This subset’s total number of permutations equals r! Because every combination of r things can be rearranged in r! different ways.

As a result, (nCr * r!) is the total number of permutations of n different items taken r at a time. It’s all about nPr.

nPr= nCr*r!

nCr =nPr/r! = n!/(n-r)!r! 

Types of combination 

We have two types of combinations- 

1. when repetition is allowed.
2. when repetition is not allowed. 

Examples of combinations based on types 

Type 1- when repetition is allowed 

Three types of dress are available in a shop. These dresses are shirts, pants, and suits (each in different colors). A girl can choose only two of them. What will be the variations in this case?

Ans – Three types of dresses are available in the shop.

A girl can choose only two of them. She can pick one dress in two different colors. 

We can use the example of variation here. 

(2shirts), 1 shirt 1 pant, 1 pant 1 suit … etc any order we can choose irrespective of it’s the arrangement.

Conclusion 

The combination is the way of selecting items in which the order or sequence of selection doesn’t matter. Formula to calculate permutation is nPr = (n!) / (n-r)!  and formula to calculate combination is nCr = nỊ / (n-r)ỊrỊ. 6Ị(6 factorial) means Multiply 6 in decreasing order until you get 1 i.e.6*5*4*3*2*1. We use permutation when we have to we have to arrange the objects and different orders are to be counted. Use combination if a problem is for the number of ways of selecting objects and the Order of selection is not important. Consider the phrase “number of options” whenever you hear the phrase “number of combinations.” It doesn’t matter what order the objects are in when you’re picking them. XYZ and XZY, for example, are two distinct arrangements with the same selection. The number of different combinations of n different things that can be made r at a time (where r is smaller than n) = nCr = nPr/r!= n!/(n-r)r! = n!/r!(n-r)