Seven-digit codes with non-decreasing digits

Question

A sign-up code is a string of $7$ digits, each from $0$ to $9$ ; repeats are allowed and leading zeros are fine. How many codes have digits that never decrease from left to right? (For example $1122345$ and $9999999$ qualify but $2345657$ does not.)

Short answer

$11440$ . A non-decreasing code is fixed once you know how many times each of the $10$ digits appears, so count non-negative integer solutions of $x_0+\dots+x_9=7$ . By stars and bars that is $\binom{16}{7}=11440$ .

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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