Determine the number of 5-tuples of integers $(x_1, x_2, x_3, x_4, x_5)$ such that $\\$ a) $x_i \geq i$ for $1 \leq i \leq 5$ ; $\\$ b) $\sum ^{5}_{i=1}x_{i}=25$ $\\$

$\\-$ This is a question from COMC 2018, I like this question a lot, hope you guys find it interesting

The answer is 1001.

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[The key is to summarize the information given and apply a proper counting method to it] $\\$ Looking at $x_i \geq i$ for $1 \leq i \leq 5$ $\\$ Since the value for $i$ is inderterminate, we reaarange the inequality $x_i \geq i \Rightarrow x_i - i \geq 0$ so it applies to all $x_i$ [Simple yet effective] $\\$ Thus we can apply $y_i$ on the second equation: $\sum ^{5}_{i=1}y_{i}=25 - (1+2+3+4+5) = 10$ $\\$ $.\\$

Number of 5-tuples for $x_i$ = Number of 5-tuples for $y_i$$\\$ Use Stars and Bars Method $\\$ Usefour barsto divide10 "1"sinto5 groupto be distributed to $x_{1 to 5}$ $\\$ Now a simple $14C4 = 1001$ would do the job.