Name: Beth
Who are you: Parent (Middle)

How many different three-digit numbers can you make using the numbers 2, 5, 7, 8 using the digits any number of times?

Hi Beth,

I have two responses for you.

How many different one-digit numbers can you make using the numbers 2, 5, 7, 8 using the digits any number of times?

That's easy, there are four. They are 2, 5, 7 and 8.

How many ways can you extend these to two-digit numbers?

Each of the four one-digit numbers can be extended to a two-digit number by appending one of the one-digit numbers. Thus you get

22, 25, 27, 28
52, 55, 57, 58
72, 75, 77, 78 and
82, 85, 87,88

Thus each of the four one-digit numbers can be extended to a two-digit number in four ways. Thus there are

4 × 4 = 16 possible two-digit numbers.

How many ways can you extend these to three-digit numbers?

Penny

and

Hi Beth.

Pick a number: 2, 5, 7, or eight.

How many choices do you have? Four.

Now for each of those four, pick each number from the list 2, 5, 7, 8.
How many pairings do you have for each of the original choices? Four
again. That means 4 x 4 ordered pairs.

Repeat the process one more time. You have sixteen ordered pairs now.
For each pair, pick each of the four digits again. That makes 16 x 4
ordered triples.

An ordered triple is a three digit number, so we've solved our question.

Hope this helps,
Stephen La Rocque.