How many computers can be labeled using a 3 digit number without repeating a number? with repeating a number?

Question

This is part of my math assignment on counting techniques and simple probabilities. Hmmm... not so simple for me :(

The question says without repeating a digit (not number) Does that make a difference, because I'm getting different answers?

Answer 1

Well, assuming you admit the degenerate cases of 3-digit numbers beginning with 0 (i.e. 000, 001, 002... etc. are valid assignments), you can label 1000 computers without repeating a number, as there are 1000 3-digit numbers (10 possibilities for the first digit * 10 possibilities for the second digit * 10 possibilities for the third). If you do not admit such cases, then only 900 may be labeled, since the requirement that the first digit be nonzero means you only have 9 choices for the first digit.

Obviously, you may label infinitely many computers using a 3-digit number if you are allowed to repeat numbers -- just label them all 100, for instance.

Of course, experience with these sort of courses informs me that you're probably not relating the problem completely accurately, and the first question is more like "how many computers can be assigned distinct labels if each label is a 3-digit number with no repeating digits?" In this case, if the degenerate case of numbers starting with 0 is allowed, we still have 10 possibilities for the first digit, but only 9 possibilities for the second (since it can't be the same as the first), and 8 possibilities for the third (since it can't be the same as any of the first 2), for a total number of 10*9*8 = 720 possible labels. If the three-digit number may not start with 0, then there are only 9*9*8 = 648 possibilities.

Likewise, if your second question is how many computers can be assigned distinct labels if the labels may contain repeating digits, then answer is either 1000 or 900, by the reasoning in the first paragraph.

Edit: yes it makes a difference. In fact, the fact that it makes a difference was pretty much the point of the second half of my post.

Answer 2

000 --- 999, that is, 1000 computers without repeating a number.

2000 computers with repeating one number. Because each of 000 - 999 can be used to label two computers.

Answer 3

with repeating:
if you can have numbers like 000, 001, 998, 999
10*10*10 = 1000
without repeating:
if you can only have numbers like 123, 243
it's 10*9*8= 720