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The number of (at most) $n-1$ distinct integers $1\le x_1,\dots,x_k\le p$ is at most $n^kp^{n-1}$

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Let $k>n\ge1$ be integers.

I have to estimate the number of choices for the numbers $x_1,\dots,x_k$, supposing that $1\le x_j\le p$, for an integer $p$ and that there are at most $n-1$ distinct numbers between the $x_j$'s.

So, let's count: having at most $n-1$ distinct numbers, imply that I have to choose at most $n-1$ values between $1$ and $p$. Thus $p^{n-1}$.

Next I have to count all the different choices I can take for my $x_j$'s: therefore I have to count in how many ways I can assign $n-1$ values to $k$ objects: this should be the number of positive integer solutions $(y_1,\dots,y_{n-1})$ to the equation $y_1+\cdots+y_{n-1}=k$.

This is the missing link between me and the estimate written in the book, which is $n^kp^{n-1}$.

Can somebody help me please? Thanks


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