Question

How many integers from 100 to 999 must be picked in order to ensure that at least two of them have a digit in common?

Answer 1

350 digits are there from 100 to 999

Answer 2

Concept introduction:

The pigeonhole principle, one of mathematics' most basic yet useful concepts, can help us in this situation. The simplest form states that if ($N+1$) pigeons fill N holes, then at least $2$ pigeons must be present in at least one hole. As a result, if $5$ pigeons occupy $4$ holes, at least one of those holes must have at least $2$ pigeons in it.

Explanation:

Given that, integers from $100$ to $999$.

We have to find, integers where at least two of them have a digit in common.

According to the question,

Each integer that was picked will occupy at least one digit from 1 to 9.

By Pigeonhole principle, picking only $9$ integers may not cause at least two of them to have a digit in common.

Final Answer:

We have to pick at least  $9$ integers to ensure that at least two of them have a digit in common.

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