# How many integers from 100 through 999 must you pick in order to be sure that at least two of...

## Question:

How many integers from 100 through 999 must you pick in order to be sure that at least two of them have a digit in common? (for example, 256 and 530 have the digit 5 in common.)

## Solving Algebra Word Problems in One Variable

We can apply a simple three-step process to solve algebra word problems.

- Identify the unknown and assign a variable to the unknown
- Formulate the equation that relates the unknown to the information provided in the question
- Solve the equation to find the unknown

Sometimes the information in the question is provided explicitly e.g. How many 3-digit integers ...?

Sometimes the information in the question is implicit and needs some thinking on your part e.g. How many integers from 100 to 999 ...?

## Answer and Explanation:

Integers between 100 and 999 have exactly 3 digits.

There are exactly 10 unique single-digit integers (0,1,2,3,4,5,6,7,8,9).

Let {eq}x{/eq} be the number of 3-digit numbers that we can form without reusing a digit, then:

$$x=\frac {\text{# of unique single-digit integers}}{\text{# digits per integer}}\\ \therefore x=\frac{10}{3}\\ \therefore x=3\frac {1}{3}$$.

Since we are interested only in whole integers, we conclude that it is possible to pick three 3-digit integers without reusing a digit. Therefore, in order to be sure that at least two 3 digit integers have a digit in common, **we need to pick four 3-digit integers**.

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