For given $N$, $x$ and $k$ such that $0\leq x \begin{align}
& \frac{(N+1-p)(N-p)}{2}\leq \frac{(N+1-x-k)(N-x-k)}{2}+\frac{(x+1)x}{2}<\frac{(N+1-p)(N-p)}{2}+\frac{p(p-1)}{2}+p+1-x
\end{align}
Binomial coefficient intervals (inequality)
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inequality
binomial-coefficients
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1Since all your binomial coefficients have $2$ as lower index, I suggest that you use $\binom n2=\frac{n(n-1)}2$ and reformulate your question as one of purely algebraic inequalities. – 2012-09-07
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0seems that $p=k+x$ makes the trick. – 2012-09-07