# What type of question should we use $\binom{n + k - 1}{k - 1}$ and others

I know there are some questions with solutions of the form

$\binom{n + k - 1}{k - 1}$

There are also questions with solution of the form

$\binom{n + k - 1}{k}$

and there are questions with solution of this form

$\binom{n + k}{k}$

I am trying to study for exam, and I am confused sometime with form of answer correspond to which type of question.

Thank you a lot!

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These aren't always associated with cerain types. You may want to review coursebook materials which show you how they arrived at the binomial coefficients. A tiny wording difference in the question may yield different binomial coefficients. – Frenzy Li Oct 19 '12 at 2:29

I would say, in fact, that all three of the expressions that you list are essentially the same and are useful in solving the same kinds of problems. $\binom{n+k-1}{k-1}$ becomes $\binom{n+k}k$ when you substitute $k$ for $k-1$, and the latter becomes $\binom{n+k-1}k$ when you substitute $n-1$ for $n$. Here’s an example: $\binom{n+k-1}{k-1}$ is the number of solutions to the equation $x_1+x_2+\ldots+x_k=n$ in non-negative integers. $\binom{n+k}k$ is the number of solutions to the equation $x_0+x_1+\ldots+x_k=n$ in non-negative integers. But these are exactly the same kind of problem: the only difference is that in one problem I had $k$ unknowns numbered $1,\dots,k$, and in the other I had $k+1$ unknowns numbered $0,\dots,k$. Thus, it’s impossible usefully to associate any of these three binomial coefficients with a particular type of problem.