# Given 2n objects, number of ways to select n objects

• nowimpsbball
In summary, the number of ways to select n objects out of 2N objects, where N objects are identical and N objects are distinct from each other, is 2^n.
nowimpsbball

## Homework Statement

Given N identical objects and N additional objects that are different from these and from each other, find the number of ways to select n objects out of these 2N objects.

## Homework Equations

Either P(n,k) or C(n,k) or n^k or maybe even (n+k+1)/k

## The Attempt at a Solution

Looking at these...so half the set is identical and the other half is distinct from the first half and from them selves. So I will call the first set S and the second set T to help me keep them apart.
So for the first set everything is identical and w can select up to N of them... my notes seem to indicate i would go about it as (S+N+1)/N and then the second set they are distinct from each other and we can select N of them, so T^N? Are those two formulas in the right direction? If so would I multiply those together to get the correct answer?

Just count them. Number of ways to select n of A, 0 of B is 1.
n-1 of A, 1 of B is n
n-2 of A, 2 of B is n(n-1), etc.

So you get sum{i=0..n}(n choose i) = 2^n.

## 1. What is the formula for calculating the number of ways to select n objects from a group of 2n objects?

The formula for calculating the number of ways to select n objects from a group of 2n objects is nCr = (2n)! / (n! * (2n-n)!), where nCr represents the combination formula.

## 2. Can this formula be applied to any number of objects?

Yes, this formula can be applied to any number of objects as long as you know the total number of objects and the number of objects you want to select.

## 3. How does this formula account for different arrangements of the same objects?

This formula takes into account all possible arrangements of the selected objects by using the combination formula, which does not consider the order of the objects.

## 4. Can this formula be used for both ordered and unordered selections?

No, this formula is specifically for unordered selections. If you want to calculate the number of ways to select n objects in a specific order, you would use the permutation formula.

## 5. Can this formula be used for selecting objects with replacement?

No, this formula is only applicable for selecting objects without replacement. If you want to select objects with replacement, you would use the combination with repetition formula.

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