number of ways of selecting $ n$ things from $3n$ things

Question

What are the number of ways of selecting $n$ things out of $3n$ things out of which $n$ are of one kind and $n$ are are second type and rest are unlike ?

Answer 1

Suppose $k$ objects have been chosen out of the $n$ dissimilar objects.

Then we have to choose $n-k$ objects from out of the two other sets. The choices would go like $0$ from the first set and $(n-k)$ from the other, or $1$ from set $1$ and $(n-k-1)$ from the other etc. This gives $n-k+1$ choices.

So we are looking for $$\sum_{k=0}^{n}(n-k+1)\binom{n}{k} = \sum_{k=0}^{n}(n+1)\binom{n}{k} -\sum_{k=0}^{n}k\binom{n}{k}$$ The first sum is just $(n+1)2^n$. The second sum $$\sum_{k=0}^{n}k\binom{n}{k} = \sum_{k=0}^{n}(n-k)\binom{n}{k} = \frac{1}{2}\sum_{k=0}^{n}n\binom{n}{k} = n\times 2^{n-1}$$ Hence the required number of ways are $(n+2)2^{n-1}$. Hope it helps you.