# 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.

1. Identify the unknown and assign a variable to the unknown
2. Formulate the equation that relates the unknown to the information provided in the question
3. 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 ...?

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. 