# How many ways? #1 (3-digit numbers)

May 18, 2006 pm31 1:28 pm

Today's question is not supposed to be a real poser: How many three digit numbers are there?

The **question about the question** is more interesting: How many different ways (different methods) can we find that lead to the answer?

6 Comments
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I can only think of two general classes of methods.

1. combinatorics: 9*10*10 = 900

2. counting: 999-99 = 900 (largest 3-digit number minus smallest) or 9*100 = 900 (100 numbers in each of nine centuries)

Are there other ways?

In the big picture sense, there are the two you named, and one you didn’t, but… I’d count your response as three (inside parentheses).

The word “different” is open to interpretation, open enough that I have had classes claim four or five methods. I’ll come back with some in couple of days.

When I said “largest 3-digit number minus smallest” I should have been more clear. I intended something like “largest 3-digit number minus the number of less-than-3-digit numbers” or maybe “largest 3-digit number minus largest 2-digit number”.

I suppose that it is possible to generate more “different” methods if you allow them to become more contraptionary . For example, you could use combinatorics but allow the possibility of zero in the first digit place and then subtract the number of first-digit-is-zero numbers: (10*10*10) – (1*10*10) = 900. But I’m curious to hear about another general class of solutions to this kind of problem!

One method that school children use is to generate a pattern:

Digits Numbers

1 9

2 90

3 N

Of course they need to go back and verify.

How many three digit combinations are there using 1,2,3,4,5,6,7,8,9,0. Any combo like 549, 769, 444 etc. Also using 0 as the first digit.

Like a license plate 250 JOS?

Thanks I would appriciate your input.

Annemarie

Tell you what, you can figure this out fairly easily.

What if only 0 and 1 were allowed? Could you make a list of all the possibilities? I bet you could. And you would quickly see where the answer came from.

The neat part is next. You could reapply what you just figured out (yourself) and solve the problem (yourself). People are often amazed by how much “hard” math they can do.

Good luck, and write back if you get stuck.