What is distinct permutation?

Answer 1

Distinct permutations are the unique permutations of n objects taking r at a time when some of the objects are repeated. For a complete example read below.

First, you need to understand what permutations are. Suppose you have 3 balls, 2 of them blue and one red. Lets call the first blue ball, B1, second blue ball, B2, and the third ball (the red one) R1. Permutation of the three balls taking the three at a time is all the different ways you can arrange those three balls in a row. That number is 3! = 3x2x1 = 6. The arrangements are listed below.

1) B1,B2,R1

2) B1,R1,B2

3) B2,B1,R1

4) B2,R1,B1

5) R1,B1,B2

6) R1,B2,B1

Now, suppose you remove the subscripts. The list looks like this:

1) B,B,R

2) B,R,B

3) B,B,R

4) B,R,B

5) R,B,B

6) R,B,B

Note that you can't differentiate between 1 and 3, they looks like the same. Also, 5 and 6 looks the same, and 2 and 4 looks the same. So the only unique combinations seems to be 1,2 and 5. Those are call the distinct permutations.

To calculate the distinct permutations of n elements taking r at a time use the formula: nPr / (k1!*k2!*...*kn!)

In the example above:

nPr = 3! = 6 -> All permutations of n elements taking r at a time

k1 = 2 = 2 -> because we have two identical blue balls

k2 = 1 = 1 -> we only have one red ball.

So, distinct permutations = 3! / (2!*1!) = 6 /(2*1) = 3. Those distinct permutations are: BBR, BRB, RBB.

You can use a software, such as Wolfram Mathematica, to generate the distinct permutations using the function DistinctPermutations.