Permutations with Repetition and Repeated Elements

In summary, the theory states that if we have x objects containing a repeats of one element and b repeats of another, the total number of permutations without repetition is equal to x!/(a!b!). When repetitions are allowed, the number of permutations becomes xx/(a!b!). This information can be found in more detail on websites such as Wikipedia or Wikibooks.
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The theory says that if you have x objects containing a repeats of one element and b repeats of another, Np(without repetition)=x!/(a!b!).

If you have x objects and repetitions are allowed, Np(with repetition)=xx, correct?

Combining these, if we have x objects containing a repeats of one element and b repeats of another, and repetitions are allowed, Np(with repetition)=xx/(a!b!). Is this right?
 
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1. What is the difference between permutations with repetition and permutations with repeated elements?

Permutations with repetition refer to arrangements of objects where repetition is allowed, meaning an object can appear more than once in a single arrangement. On the other hand, permutations with repeated elements refer to arrangements where the objects themselves are repeated, but each individual object can only appear once in a single arrangement.

2. How do you calculate the number of permutations with repetition?

The number of permutations with repetition can be calculated by taking the total number of objects, raising it to the power of the number of slots or positions available, and accounting for any repeated objects by dividing by the factorial of the number of times each object is repeated.

3. Can permutations with repetition be used in real-world scenarios?

Yes, permutations with repetition can be used in real-world scenarios such as creating passwords with repeated characters, designing patterns on fabrics or tiles, and arranging objects or people in a specific order for events or performances.

4. What is an example of a permutation with repeated elements?

An example of a permutation with repeated elements could be arranging the letters in the word "MISSISSIPPI" in different orders. In this case, the objects (letters) are repeated, but each individual letter can only appear once in a single arrangement.

5. Are there any limitations to using permutations with repetition and repeated elements?

One limitation of using permutations with repetition and repeated elements is that the number of possible arrangements can become very large very quickly. This can make it difficult to calculate and work with in situations with a large number of objects or repetitions.

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