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Number of ways to pick $n$ pairs of objects from $n+1$ objects (the figures can be the same)
I’m trying to solve a problem in my textbook, it’s quite simple but I just don’t know how to approach it.
What is the number of ways to pick $n$ pairs of objects from a set of $n+1$ objects (the figures can be the same)?
For example, there are $4$ different pairs of objects from set of $5$ objects.
I tried to count the amount of ways $5$ can be chosen, i.e. $4$ ways for the first object, $4$ ways for the second object, $3$ ways for the third object, etc. and sum it up, but I don’t think I can simplify the calculation so much.
Any ideas? Thanks.
A:
HINT: There are $1$ pairs of sets, and there are $\binom{n+1}2$ ways to choose $n$ of them. That means there are $\binom{n+1}2$ distinct ways for two sets to be paired. You want to determine the number of ways in which the remaining $n$ sets can be chosen in distinct ways, and there are $\binom{n+1}3$ ways to do that, so the answer is
$$\binom{n+1}3-\binom{n+1}2=n+1\binom{n+1}2-\binom{n+1}2=n\binom{n+1}2$$
A:
Each pair of objects is equally likely to be picked, so there are $1$ pair of objects. This leaves $n+1$ objects, so there are $\binom{n+1}{2}$ pairs. However, since the chosen pair must be distinct, there are $\binom{n+1}2$ ways to select the distinct pairs.
Q:
How to assign the program output to a list in python
I’ve been working on this for
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