Help with permutations and combinations

In summary, to calculate the total number of combinations when choosing r objects from a group of n objects, subtract the number of combinations where one of the mutually exclusive events has occurred from the total number of permutations. This can be represented by the formula \binom{n}{k}-\sum_{i=2}^{x}\left[\binom{x}{i}\times\binom{n-x}{k-i}\right] or, for x=2, \binom{n}{k}-\binom{n-2}{k-2}.
  • #1
Ryoukomaru
55
0
How do u calculate the the total number of combinations, given that you have n number of object and you will choose r of the objects, but x of these objects are mutually exclusive. Let x=2 for your explanations.

I kinda have an idea on how to do this, but i can't frecall an formula for the calculations.

Total number of permutations would be [tex]\frac{n!}{(n-r)!r!}[/tex] And now i have to subtract from this, the number of combinations with one of the mutually exclusive events given that the other has happened.

P.S. Maybe i should post a question to help you understand better ? ..., as my linguistic skills are not top-notch.
 
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  • #2
If the objects are mutually exclusive that means you can only select 1 of them. So you choose from n - x + 1 objects, and multiply it by x ways to choose from the x objects.
 
  • #3
I don't think that's quite right. It ignores the combinations that don't have one of the x objects.
 
  • #4
A solution for this problem is:

[tex]\binom{n}{k}-\sum_{i=2}^{x}\left[\binom{x}{i}\times\binom{n-x}{k-i}\right][/tex]

I'm sure there's a more elegant formulation, but this one works.

Of course, for x=2, this simplifies to:

[tex]\binom{n}{k}-\binom{n-2}{k-2}[/tex]
 
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1. What is the difference between permutations and combinations?

Permutations and combinations are two different mathematical concepts that involve selecting and arranging items from a larger set. The main difference is that permutations consider the order of the items, while combinations do not. In other words, permutations are arrangements where the order matters, while combinations are selections where the order does not matter.

2. How do I know whether to use permutations or combinations?

To determine whether to use permutations or combinations, you need to consider the problem at hand. If the order in which the items are selected or arranged matters, then you should use permutations. If the order does not matter, then combinations should be used. Additionally, the number of items being selected or arranged can also help determine which concept to use. Permutations are used when all the items are being arranged, while combinations are used when a subset of items is being selected from a larger set.

3. Can you give an example of a permutation?

Yes, an example of a permutation would be the different ways you can arrange the letters in the word "CAT". There are 6 possible permutations: CAT, CTA, ACT, ATC, TCA, and TAC. Notice that the order of the letters changes in each permutation.

4. What is the formula for calculating permutations and combinations?

The formula for permutations is n! / (n - r)! where n is the total number of items and r is the number of items being arranged. The exclamation mark represents the factorial operation. The formula for combinations is n! / (r!(n-r)!), where n is the total number of items and r is the number of items being selected. Again, the exclamation mark represents the factorial operation.

5. Are there real-life applications of permutations and combinations?

Yes, permutations and combinations have many real-life applications, especially in fields such as mathematics, statistics, physics, and computer science. They are used to solve various counting problems, such as calculating the number of possible outcomes in a game, the number of possible combinations in a lock, or the number of ways a group of people can be seated at a table.

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