# Mastering Combinatorics: Exploring the Formula for Permutations

• 12john
In summary, the conversation revolved around finding the explicit formula for the number of permutations of a given set of elements with specific quantities, as well as the need to elaborate on the solution provided by the student. The student's answer was correct, but it was deemed too brief and lacking in detail. The instructor recommended seeking further clarification and explanation from the instructor for improvement.
12john
Thread moved from the technical forums to the schoolwork forums
My combinatorics professor has a MA, PhD from Princeton University. On our test, she asked

What's the explicit formula for the number of ##p## permutations of ##t## things with ##k## kinds, where ##n_1, n_2, n_3, \cdots , n_k## = the number of each kind of thing ?

I handwrote, but transcribed in Latex, my answer below.

To deduce the formula for all the unique permutations of length ##l## of ##\{n_1,n_2,...,n_k\}##, we must find all combinations ##C=\{c_1,c_2,...,c_k\}## where ##0 \leq c_k \leq n_k##, such that
##\sum_{i=1}^k c_i=l##.

What we need, is actually the product of the factorials of the elements of that combination:
##{\prod_{i=1}^k c_i!}##

Presuppose that the number of combinations is J. Then to answer your question, the number of permutations is
$$= \sum_{j=1}^J \frac{l!}{\prod_{i=1}^k c_i!} = \sum_{c_1+c_2+...+c_k=l} \binom{l}{c_1,c_2, \cdots ,c_n},$$
as a closed form expression with a Multinomial Coefficient. *QED.*

How can I improve this? What else should I've written? Professor awarded me merely 50%. She wrote

Last edited by a moderator:
12john said:
I handwrote, but transcribed in Latex, my answer below.
I edited your LaTeX. At this site we use MathJax, which has to be delimited by either pairs of # characters (for inline TeX) or pairs of \$ characters (standalone).
I also removed all the color stuff. We prefer that you use a minimum of extra color, bolding, italics, etc.
12john said:
How can I improve this? What else should I've written?
What she wrote was "You need to elaborate." The best explanation would come from your instructor.

## 1. What is combinatorics and why is it important in mathematics?

Combinatorics is a branch of mathematics that deals with counting and arranging objects. It is important because it allows us to solve problems involving permutations and combinations, which are essential in many areas such as probability, statistics, and computer science.

## 2. What is the formula for permutations?

The formula for permutations is n! / (n-r)!, where n is the total number of objects and r is the number of objects being selected and arranged in a specific order.

## 3. How do I know when to use permutations instead of combinations?

Permutations are used when the order of the objects matters, while combinations are used when the order does not matter. For example, if you are selecting a president, vice president, and treasurer from a group of 10 people, you would use permutations because the order of the positions matters. However, if you are selecting a group of 3 people to be on a committee, you would use combinations because the order of the members does not matter.

## 4. Can you provide an example of a real-life application of permutations?

One example of a real-life application of permutations is in password creation. When you create a password, you are essentially selecting and arranging characters in a specific order. The number of possible permutations for a password with 8 characters is 8!, which is over 40,000,000,000,000,000. This makes it difficult for hackers to guess your password.

## 5. Are there any shortcuts or tricks for solving permutation problems?

Yes, there are a few shortcuts or tricks that can make solving permutation problems easier. One is the use of factorial notation, which allows you to quickly calculate the number of permutations. Another is the use of the fundamental principle of counting, which states that if there are n ways to do one task and m ways to do another task, then there are n x m ways to do both tasks together. Additionally, understanding the difference between permutations and combinations can help you determine which formula to use in a given problem.

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