The two mathematical terms are often associated with one another loosely. However, not only are Permutation and Combination Formulas different, the whole concept of what they mean in the mathematical sense is also very different.
By definition, permutation is the arrangement of values of a set in a linear order. The word permutation loosely translates to the act of approaching various elements of a set in an ordered manner.
However, by mathematical definition only, an arrangement of the elements of a set where the order is irrelevant is referred to as a combination. The combination is not affected by the order in which the values are picked; instead, it is affected by the choice in which the values are picked.
Applications of permutations and combinations
In everyday life, we use permutation and combination quite often, even as we fail to comprehend the frequency at which permutations and combinations govern our day to day actions. Below are some examples of when you might have used permutations and combinations without even realising it:
- While travelling, you make plans to decide how you want to visit some cities in another state or country. If you follow through with this order, you see a permutation in action.
- Similarly, you might want to travel haphazardly while road-tripping, but you still follow a certain order. This order may not be linear but is the combination of one of the many orders you could have picked to travel those cities in. This is a fine example of combination in action.
- The permutation is used in combinationlocks. This may sound confusing, but while entering a code in a combination lock, you pick one of many possible combinations of numbers in a very particular order that can open this lock. This is a linear order from a set of numbers to choose from; thus, it is a permutation.
- Another excellent example of permutations in action in everyday life is license plates. When you buy a new vehicle, a unique license plate is assigned to your vehicle to differentiate it from other cars of the same make and model. These license plates are made from a permutation of available numbers in a set of predefined numbers that you can choose from.
- You also use combinations while picking your everyday outfit. You have an option to pick from several items of clothing and pair them together. All these possible outcomes are different combinations you can use. For instance, if you have 5 shirts and 6 pants, you can wear them in a total of 30 different combinations.
- Moreover, a good example of permutations is phone numbers. To ensurethat two people do not have the same 10 digits mobile number that can cause a lot of confusion, every sim card that gives you a number chooses it through permutation. This is useful because it ensures that a different set of numbers in a particular linear order are given to each different sim card.
- A combination is also used to pick lottery winners. It is not required to pick your lottery numbers in order but having the same combination of numbers randomly picked as winners will win you a lottery.
- Another great example of using combinations is when you are ordering food at a restaurant. There are several options you can pick from. Suppose you choose to order pizza, and there are 10 different toppings, 3 different bases and 4 sides you can pick from. Since there is no requirement to follow a set order, you can create many combinations of pizzas you could possibly order.
The permutation formula
Now that we know that permutation is a set of values from which values are picked in a linear order, we can understand the permutation formula. Simply put,
nPr = n!/(n – r)!
Where nPr represents permutations, n! is the factorial of the number of values in the set, and r is the number chosen.
The above is calculated as follows:
By the fundamental counting principle that states that if there are n ways of doing something and m ways of doing another thing, there are n x m ways of doing both the things we have,
P(n, r)= n. (n-1). (n-2)…..until (n-(r-1)
Therefore, P(n, r) =n. (n-1). (n-2)…. (n- r plus 1)
By multiplying and dividing both sides by (n – r) (n – r – 1) . . . 3 * 2 * 1, we get,
nPr =n( n -1)(n-2)….(n-r plus 1)(n-r) (n- r-1)… 3*2*1/ (n -r) (n-r-1)…. 3*2*1 = n!/(n – r)!
Therefore, nPr = n!/(n – r)!
The combination formula
Since combination is a set of values from which values are picked irrespective of any order, we can understand the formula for combination. Simply put,
nCr = n!/ r! (n – r)!
Where nCr is the representation of combinations, n! is the factorial of the number of values in the set and r! is the factorial of the number that can be chosen.
This formula can be derived from the formula of permutations as follows:
Since the total number of permutations of n different things taken r at a time is nCr × r! because r objects in every combination can be rearranged in r! ways,
Therefore, nPr = nCr × r!
This indicates that n!/(n – r)! = nCr × r!
Thus, nCr = n!/ r! (n – r)!
Solved example to understand permutations
- Calculate the permutations if n= 15 and r = 3
Since, nPr = n!/(n – r)!
=15! / (15 -3)!
= 15! / 12!
= 15 x 14 x 13 x 12! / 12!
= 15 x 14 x 13 = 2730
Solved example to understand combinations
- Calculate the combinations if n = 12 and r = 2
Since, nCr = n!/ r! (n – r)!
=12! / 2! ( 12 -2)!
= 12! / 2! (10)!
= 12 x 11 x 10! /2 x 1 x 10!
= 12 x 11 / 2
= 132 / 2
= 66
Conclusion
Permutation and combination are two different mathematical operations. They are more commonly referred to at the same time as the other, but it is important to understand where and how they differ. They are also very commonly applied in everyday life; therefore, it is essential to have a basic understanding of permutations and combinations.
