AoPS - Problem

Processing math: 100%

Problem

Source: Kosovo National Olympiad 2025, Grade 7, Problem 3

Tags: combinatorics

ItsPena

17.11.2024 19:40

A number is said to be regular if when a digit $k$ appears in that number, the digit appears exactly $k$ times. For example, the number $3133$ is a regular number because the digit $1$ appears exactly once and the digit $3$ appears exactly three times. How many regular six-digit numbers are there?

BR1F1SZ

17.11.2024 20:12

Note $6=1+2+3$ , $6=1+5$ , $6=2+4$ , $6=6$ are all the ways to express $6$ as the sum of different integers. Hence, there are $6 \times \frac{5\times4}{2}+6+\frac{6\times5}{2}+1=\boxed{82}$ regular six-digit numbers.