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# Permutations and Combinations

Prithvi Krishna
11 Jul 0 0

Suppose there are 'n' number of different things. You would like to 'select' or 'arrange' all or some of those things in an order. In how many unique ways can you do so? Permutations and combinations is the topic that deals with this.

Suppose you want to arrange 'r' number of things out of 'n' number of things(n>r). This is given by permutation nPr. This is equal to n!/(n-r)!. Now, how do we get this?  We have number of tihngs. In how many ways can we select first thing? Right, 'n' ways(because there are 'n' number of different things). Now, in how many ways can we select the second thing? 'n-1' ways(why? because, one thing has already bene selected as the first thing out of n things). Similarly, r th thing can be selected in (n-r+1) ways(why +1? think!). So, 'r' things can be collectively selected in nP= n * (n-1) * (n-2) * ... * (n-r+1) ways. Multiplying and divind the RHS by (n-r)!, we get nPr  = n!/(n-r)!

Now, asume, you are only interested in selection and not arrangement. Then, it is called a combination. This combination can be represented as nCr = n!/(r!* (n-r)!). Here, r! appears in denominator because there is no arrangement and only selection. In permtutation, we had also considered arrangement and hence r! appears in denominator in combination.

These are the basic definitions of Permutation and combination. There are several interesting problems that can be solved using this simple concept. Permutations and Combinations also are essential to solve problems related to Probability.

Hope this short lesson was useful and you found it enjoyable.

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