Permutation Calculator

    A permutation calculator is a handy tool which helps you find out how many permutations a given set has. Permutation is often symbolized as nPr which we will also use in this article. Let’s cover how to use this permutation formula calculator, how to calculate permutation without an nPr calculator, and more.

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    How to use the permutation calculator?

    This permutation calculator is simple, easy to understand, and extremely easy to use too. All you have to do is input 2 values and it will do the calculation for you.

    This online tool is just as useful as a permutations and combinations calculator which provides you with both the permutations and the combinations of a given set. Here are the steps to follow for this calculator:

    • First, enter the value of the Number of Objects (n) in the appropriate field.
    • Then enter the value of the Sample Size (r) in the appropriate field.
    • After entering both values, the permutation formula calculator automatically generates the value of the Permutations of the set for you.

    How do you calculate a permutation?

    Permutation or nPr refers to the number of ways by which you can select r elements out of any given set containing n distinct objects. Unlike combinations, when it comes to permutations, the order of selecting the elements has relevance.

    For instance, let’s assume that you have a whole deck of cards which have numbers from 1-9. Select 3 cards from the deck randomly then place them on a table in a line to create a number with 3-digits. For these three cards, how many different numbers can you come up with?

    The good news is that you don’t have to list down all of the possible 3-digit numbers in such a case. You can either use the nPr calculator or the permutation formula if you want to perform the calculation manually. Either of these allows you to calculate the number of permutations easily. Here is the formula:

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

    where:
    p refers to the number of permutations
    n refers to the total number of elements in a given set
    r refers to the number of elements you select from the given set

    The exclamation point in the equation represents a factorial. If you need to learn more about factorials or you need to perform factorial calculations, check out the factorial calculator.

    As you can see in the permutation formula, the number of permutations for when you select a single element is n. On the other hand, if you need to select all of the elements, you must modify the formula slightly. Therefore, the formula becomes:

    P(n,n)=n!.

    To help you understand this better, let’s go back to our earlier example. Let’s apply the formula to our situation with the deck of cards. In this case, you must solve for the number of ways to select 3 cards out of the total 9. Replace the numbers in the equation:

    P(9,3) = 9!/(9-3!) = 9!/6! = 504

    After performing the calculation, you can check the accuracy using the permutation calculator. Then keep using this formula every time you need to calculate permutations manually in any given situation.

    What is the formula for nPr?

    As aforementioned, permutation and nPr mean the same thing. nPr is the symbolic representation of permutation. This means that the formula for nPr is the same as the formula for permutations which is:

    P(n,r)=n!(n−r)! for n ≥ r ≥ 0

    By definition, permutations refer to how many ways you can get n ordered subsets of elements r from a set of elements n. Permutations differ from combinations because of the significance of the order of the elements. The permutation of n elements taken r at a time also means the number of ways that you can order r objects you selected from distinct n objects in a set.

    In mathematics, it’s denoted as the r at the bottom-right corner and n at the top-left corner of P. This is why another common form of notation of permutation is P(n,r) as seen in the permutation formula. When you use the basic principle of counting, you will be more familiar with the symbol P(n,r) which is a closed expression.

    Let’s have another example to help you understand the nPr formula further. Let’s assume that you have r places. For the first place, you can fill it by selecting any of the n elements and putting them in this place. For these elements in the first place, you can arrange them in n ways.

    Since you’ve already filled the first place, you now have n-1 elements remaining. Now you can select elements from any of the n-1 elements left for the second place. Again, you can arrange the elements in the second place in n number of ways.

    Keep going until you’ve filled the nth place or until you run out of elements. In doing this, you will observe something interesting:

    For the first place, n=n−(1–1) ways
    For the second place, n−1=n−(2–1) ways
    For the third place: n−2=n−(3–1) ways, and so on

    As you can see, this progression suggests that you can fill up the rth place in n−(r−1)=n−r+1 ways.

    Since these events occur all at the same time or simultaneously, the total number of arrangements possible is the product of how many ways there are to fill each individual r place. To close this expression, multiply then divide it by (n-r)!. Obviously, the numerator then becomes n!. This is why the equation for permutation is:

    P(n,r)=n!(n−r)!

    What is the difference between nCr and nPr?

    While both nCr and nPr are quantities in statistics which refer to possible subsets of a given set of objects, they aren’t totally identical. First, they have different formulas which are:

    nCr(n,r)=nPr(n,r)/r!=n!/(r!(n−r)!)
    nPr(n,r)=n!/(n−r)!

    When you use sets to define these statistical quantities, nPr refers to the formula to solve for permutations of n elements which you take r at a time. Here, the arrangement or order of the elements matters. For nCr, this refers to combinations wherein you don’t take into consideration the arrangement or order of the elements.