 

permutation & combination calculator  step by step calculation to find number of different permutations nPr & combinations nCr provided along with meaning, formula & solved example problems for statistics data analysis. Users can get the complete step by step calculation for each calculation they do by using this calculator. nPr  permutation calculator finds the number of different permutations of n distinct objects taken r at a time where as the nCr calculator finds the number of different combinations of n distinct objects taken r at a time. The factorial of number calculation one of the primary part in both permutation & combination calculation.
The below mathematical formulas are used to find the different number of permutations P(n,r) & combinations C(n,r) of n distinct objects taken r at a time
The below solved example problem may useful to understand how the values are being used in permutations P(n,r) & combinations C(n,r) calculation by using the above formulas.
Example Problem
Find the number of different permutations nPr & combinations nCr of a box containing 6 distinct colour balls taken 3 at a time?
Solution
Data given
n = 6
r = 3
Step by step calculation
formula to find permutation nPr = n!/(nr)!
n! = 6! = 6 x 5 x 4 x 3 x 2 x 1
n! = 720
(n  r)! = 3! = 3 x 2 x 1
(n  r)! = 6
r! = 3! = 3 x 2 x 1
r! = 6
substitute the values
= 720/6
nPr = 120
formula to find permutation nCr = n!/(r!(nr)!)
substitute the above values
= 720/(6 x 6)
nCr = 20
In the context of counting problems, permutations is the arrangements where the order is important and repetitions or recurrence is not allowed. The number of different permutations of n distinct points taken r at a time is written as nPr. Because the number of objects is being arranged cannot exceed total number available. There are n! (n factorial) permutations of n distinct objects. A rpermutation of n objects is a permutation of r of them. There are n!/(n  r)! different r  permutations of n symbols. Refer the below table for example input & output of permutations calculator. It's usually represented by nPr and calculated from the below formula
nPr = n!/(n  r)!
Input  ^{n}P_{r} 

^{2}P_{1}  2 
^{3}P_{1}  3 
^{3}P_{2}  6 
^{4}P_{1}  4 
^{4}P_{2}  12 
^{4}P_{3}  24 
^{5}P_{1}  5 
^{5}P_{2}  20 
^{5}P_{3}  60 
^{5}P_{4}  120 
^{6}P_{1}  6 
^{6}P_{2}  30 
^{6}P_{3}  120 
^{6}P_{4}  360 
^{6}P_{5}  720 
The combinations is a method of selecting several items or symbols out of a larger group or a data set, where an order does not matter. Refer the below table for example input & output of nchoosek calculator. It's usually represented by nCr and calculated from the below formula
nCr = n!/(r!(n  r)!)
nCHOOSEk  Output 

2 choose 1  2 
3 choose 1  3 
3 choose 2  3 
4 choose 1  4 
4 choose 2  6 
4 choose 3  4 
5 choose 1  5 
5 choose 2  10 
5 choose 3  10 
5 choose 4  5 
6 choose 1  6 
6 choose 2  15 
6 choose 3  20 
6 choose 4  15 
6 choose 5  6 
Each r combination can be arranged in r! different ways. Then the number of rpermutations is equal to the number of r combinations times r!
Factorial is used to compute permutations (nPr) and combinations (nCr). A factorial is the result of multiplying a given number of consecutive integers from 1 to the given number. It is written with the exclamation sign: n! and it is defined as
0! = 1
1! = 1
2! = 2 x 1 = 2
3! = 3 x 2 x 1 = 6
4! = 4 x 3 x 2 x 1 = 24
5! = 5 x 4 x 3 x 2 x 1 = 120 and so on.
In many applications in the field of probability & statistics, finding the permutations & combinations is very important to analyse and summarize the statistical data. The above formulas, step by step calculation & example solved problem help users to understand how the values are being used in nPr & nCr calculations and how to be done such calculations manually but, when it comes to online for quick computations, this permutation & combination calculator help users to workout, perform & verify such calculations as quick as possible.
Similar Resource 
Worksheet for how to Calculate Permutations nPr and Combination nCr 