Ordered triples of positive integers summing to 25

Question

In how many ways can $25$ be written as $x_1+x_2+x_3$ with each $x_i$ a positive integer, where the order of the terms matters?

Short answer

$276$ . Subtract $1$ from each term to get non-negative integers summing to $22$ ; by stars and bars there are $\binom{24}{2}=276$ solutions.

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Lessons behind this question

Counting and Combinatorics
Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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