Another how many combinations question - slightly more complicated

[krytie75]

Hi. I need a little help.

I'm trying to work out how many combinations of characters there are for a string that is between 1 and 8 characters long and uses a-z (non caps) and 0-9. There are 36 useable characters and the string could be anything from simply 'a' to '99999999'. I was able to work out how many combinations there would be if there were known to be 8 characters in the string, but I'm not sure how to include all the combinations before that (1 - 7 characters long). Is there a formula for this?

Thanks in advance!

 

[Mentallic]

Hi. I need a little help.

I'm trying to work out how many combinations of characters there are for a string that is between 1 and 8 characters long and uses a-z (non caps) and 0-9. There are 36 useable characters and the string could be anything from simply 'a' to '99999999'. I was able to work out how many combinations there would be if there were known to be 8 characters in the string, but I'm not sure how to include all the combinations before that (1 - 7 characters long). Is there a formula for this?

Thanks in advance!

It's actually simpler than you think.

How many combinations are there if it's just 1 character length? 36
Two characters in length? Well the first can be any of the 36, and the second can be any of 36 as well, so it's 36²
Three characters? Again applying the same idea, we get 36³
...

And so we can get the total number of combinations by adding each and every value, so we have
$$36+36^2+36^3+...+36^8$$

And just as a reminder, we can evaluate this more simply by using the formula $$1+r+r^2+...+r^n=\frac{1-r^{n+1}}{1-r}$$

 

[krytie75]

Hi Mentallic!

Thanks for your help, it all makes perfect sense!

One thing I'm not quite sure on though...

When I use the formula you provided at the bottom of your response, I get the answer:

2901713047669

However when I manually calculate 36+36^2+...+36^8 I get the answer:

2901713047668

Which has a difference of 1. I'm guessing this has something to do with the "1+r+r^2" bit of your formula but I'm not sure why?

Thanks again!

 

[Mentallic]

If you use the formula, you should get 2901713047668 as the answer should be. Honestly, I can't quite think where it went wrong for you so you'll have to show me what your procedure was.

edit: If I was to take a guess, you added 1 in front of the long expression to get $1+36+36^2+...+36^8$ to get it in the form $1+r+r^2+...+r^n$ in which you then used the formula and plugged in your values as $$\frac{1-36^9}{1-36}$$ and got the higher wrong answer because you forgot to take that 1 you added back out of the value.

Also, an easier way to get it into the form of that formula would be to factorize out 36 so you're then solving $$36(1+36+36^2+...+36^7)=36\cdot \frac{1-36^8}{1-36}$$

 

[krytie75]

= (1-36^8+1)/(1-36)

= (1-36^9)/(1-36)

= -101559956668415/-35

= 2901713047669

Sorry, I'm new here and don't know how to use the proper maths characters yet.

 

[Mentallic]

= (1-36^8+1)/(1-36)

= (1-36^9)/(1-36)

= -101559956668415/-35

= 2901713047669

Sorry, I'm new here and don't know how to use the proper maths characters yet.

Yep, there you go, that's why.

Well you're doing a lot better than many other new guys that come to this forum - you know how to use parenthesis properly!

 

[krytie75]

Ahhh! I forgot my basic maths and didn't balance the equation.

Thanks Mentallic, that's really appreciated.

 

[Diffy]

If the String Can be "" That is a string of length 0 with nothing in it then there are 2,901,713,047,669 possibilities. However you did say that it had to be of at least length one which means that "" is not a valid string combination you want to count.
Thus 2,901,713,047,668 would be your correct answer.

Just thought I would add some context to the 1 difference you were seeing between the formulas.