# Phone Call

How many different phone numbers are there starting with a 3 and with at most two different digits?

## Problem

A new taxi firm needs a memorable phone number.

They want a number which has a maximum of two different digits.

Their phone number must start with the digit $3$ and be six digits long.

How many such numbers are possible?

If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.

## Student Solutions

There is the possibility of using only $3$s giving one possible number $333333$.

Let's suppose a second digit is used, say $x$. After the initial digit $3$, there are five positions into which we can put either $3$ or $x$. So there are two choices in each of these five positions and so $2^5 = 32$ possible choices - except that one such choice would be five $3$s. So we get $31$ choices.

There are nine possible values for $x$, namely $0$, $1$, $2$, $4$, $5$, $6$, $7$, $8$, $9$.

So this gives $9\times 31=279$ numbers.

Together with $333333$, this gives $280$ numbers.