Combinatorics clarification?

In summary, there are 5^7 ways to assign 5 grades to 7 people, as each person can have 5 choices of grades and the choices do not affect each other. This is determined by multiplying the number of choices for each person, which is 5, for a total of 7 people.
  • #1
MathWarrior
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5
Given 5 grades how many different ways are there to assign them to 7 people?

How do you determine that this should be [itex]5^{7}[/itex] and not [itex]7^{5}[/itex]?
 
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  • #2
How many ways are there to assign 5 grades to the first person? Clearly 5.
For two people, you fix the grade of the first person, and give the 5 grades to the next person, and then change the grade of the first person, and again give the 5 grades to the next person. Do this for all 5 grades of the first person. That's essentially 5*5 = 52 ways to give grades to 2 people.
When you have a 3rd person, similarly you give him a grade and then go through all the possible ways to give a grade to the other two people, which we know is 52 ways, and then repeat this process for all 5 grades on that 3rd person. This is 5*52=53.

etc...
 
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  • #3
Similar to what Mentallic has said. The first person can have 5 different choices of grades, and whatever grades that were assigned to the first person will not affect the number of choices available for the second person.
So, the second person will also have 5 choices of grades. The third person and so on will follow in the same manner.
Hence, you will get 5 x 5 x 5 x 5 x 5 x 5 x 5 = 5^7 ways
 

1. What is combinatorics?

Combinatorics is a branch of mathematics that deals with counting and organizing objects or arrangements. It involves studying the different ways in which a collection of objects can be selected, arranged, or grouped together in a specific order.

2. Why is combinatorics important?

Combinatorics plays a crucial role in various fields such as computer science, statistics, physics, and biology. It provides tools and techniques to solve problems related to counting, probability, and optimization. It also helps in understanding and analyzing various phenomena in the real world.

3. What are the main principles of combinatorics?

The main principles of combinatorics include permutations, combinations, and the multiplication and addition principles. Permutations deal with the arrangement of objects in a specific order, combinations deal with the selection of objects without considering the order, and the multiplication and addition principles are used to calculate the total number of outcomes in a given scenario.

4. What are some common applications of combinatorics?

Combinatorics has many practical applications, such as in cryptography, scheduling, network design, and genetics. It is also used in various industries, including finance, marketing, and manufacturing, to make decisions and solve problems.

5. What are some common misconceptions about combinatorics?

One common misconception about combinatorics is that it only involves counting and is not a significant branch of mathematics. In reality, combinatorics is a versatile and powerful tool that can be applied to various real-world problems. Another misconception is that combinatorics is only useful in theoretical scenarios, but it has many practical applications in different fields.

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