2018 COMC Problem #B4

[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 $\\$ Use four bars to divide 10 "1"s into 5 group to be distributed to $x_{1 to 5}$ $\\$ Now a simple $14C4 = 1001$ would do the job.