Counting solutions with a sum between 6 and 10

Question

How many $5$ -tuples $(x_1,\dots,x_5)$ of non-negative integers satisfy $6\le x_1+x_2+x_3+x_4+x_5\le 10$ ?

Short answer

$2751$ . Adding a slack variable, the number of tuples with sum at most $m$ is $\binom{m+5}{5}$ . So the count is $\binom{15}{5}-\binom{10}{5}=3003-252$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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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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