[JEE Advanced 1996] For $n\ge\frac{k(k+1)}2$, the number of integer solutions $(x_1,\dots,x_k)$ with $x_i\ge i$ and $x_1+\cdots+x_k=n$ is ____
For $n\ge\frac{k(k+1)}2$, the number of integer solutions $(x_1,\dots,x_k)$ with $x_i\ge i$ and $x_1+\cdots+x_k=n$ is ____
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Answer: $\dbinom{\,n-\frac{k(k+1)}2+k-1\,}{\,k-1\,}$.
Put $y_i=x_i-i\ge0$; then $\sum y_i=n-\frac{k(k+1)}2$, and the number of non-negative solutions is $\binom{n-\frac{k(k+1)}2+k-1}{k-1}$.
JEE Advanced 1996 · Quadratic Equations and Inequations — verified solution by the Vidaara Team.
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