Phone call
How many different phone numbers are there starting with a 3 and with at most two different digits?
Age
14 to 16
Challenge level
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.