Permutation with Repetitions Allowed

There are many different types of permutations, but today we’re going to focus on permutations with repetitions allowed. This type of permutation is particularly useful when you need to find all the possible solutions to a problem. For example, if you were asked to create a word using the letters A, B, C, D, and E, you could use the following method: ABBCDE, ABCDEE, ACBDEA etc. As you can see, by allowing repetitions, the number of possible solutions increases exponentially! In this article, we will discuss how to find permutations with repetitions allowed and provide some examples for clarification.

What Are Permutations?

A permutation is an arrangement of objects in which order matters. For example, if you have a pool of six balls—numbered one through six—a permutation would be any set of those balls put together in a particular order: {ball number one, ball number two, ball number three}, or {ball number four, ball number six, ball number five}.

There are two types of permutations: with and without repetition. In our example above, if balls one, two and three are put together in that order, it’s called a permutation with repetition because the same numbers can appear more than once. If we take those same six balls but now they can only be used once, it’s called a permutation without repetition. So, a permutation without repetition of those six balls might be {ball number one, ball number four, ball number six}.

Permutations with repetitions are often referred to as “with replacement.” That’s because the same item can be used more than once. For example, if you have a bag of five candy bars and you want to find out how many different ways you can choose three candy bars, you would use permutations with repetitions.

To calculate the number of permutations with repetition, we use the following formula: nr

“n” is the number of items to choose from, and “r” is the number of items we want to choose. In our candy bar example, we have five candy bars to choose from (n=5 and r=3)

The Best Way to Find the Solution

If you’re looking for the best way to find the solution, you’ve come to the right place. We’re going to show you how to use permutations with repetitions to find the answer you’re looking for.

First, let’s take a look at an example. Let’s say you have a group of four people and you want to know how many different ways you can choose two of them. In other words, you want to know how many permutations with repetitions there are.

To calculate this, we would use the formula we mentioned earlier: nr

“n” is the number of items to choose from, and “r” is the number of items we want to choose. In this case, we have four people to choose from (n=4 and r=2)

Therefore 4²=16

That means there are a total of 16 different permutations with repetitions.

Examples of Permutations With Repetition

Now that we’ve seen how it’s done, let’s try it ourselves.

Let’s say you have a group of four people: John, Paul, George, and Ringo.

You could list all of the possible ways that these four people could sit in four chairs like this:

John, Paul, George, Ringo

John, Paul, Ringo, George

John, George, Paul, Ringo

John, George, Ringo, Paul

John, Ringo, Paul, George

John, Ringo, George, Paul… and so on!

We can simply apply the formula n^r, which means 4^4= 265.

Conclusion

Permutation with repetitions allowed is a mathematical operation that calculates all the possible orderings of a set of objects. In other words, it determines all the different ways you can arrange a group of things. For example, if you have four items {A, B, C, D}, there are six permutations with repetitions allowed: ABCD, ADBC, BCDA, BDCA, CDBA and CBAD. If you have eight items {A-H}, there are 28 permutations with repetitions allowed: AABBCCDDEEFFGGHHIIJJKKLLMMNNOOOPPPQRRSSTTTUVWXYZ. The number of permutations with repetitions allowed will always be equal to nr, where n is the number of items and r is the number of repetitions.