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Each user on a computer system has a password, which is six to eight characters long, where each character is an uppercase letter or a digit. Each password must contain at least one digit. How many possible passwords are there?

P6 = $36^{6} − 26^{6}$ = 2,176,782,336 − 308,915,776 = 1,867,866,560.

Similarly, we have

P7 = $36^{7} − 26^{7}$ = 78,364,164,096 − 8,031,810,176 = 70,332,353,920

and P8 = $36^{8} − 26^{8}$ = 2,821,109,907,456 − 208,827,064,576 = 2,612,282,842,880.

Consequently,

P = P6 + P7 + P8 = 2,684,483,063,360.

My question is, instead of using the technique to find out P6 above, is there a slower way to do so without using subtracting full values? I ask this mainly for the purpose of solidifying my understanding of counting; in practice, I would prefer the technique above. thanks!!

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    If you are asking for the purpose of solidifying your understanding, shouldn't you be asking for smarter, easier ways to count? Why do you care about more stupid, harder ways? – 2017-01-20
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    You want to simplify calculations?? – 2017-01-20
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    @ mathguy The smartest way doesn't necessarily allow me to look under the hook, for example, you can follow the best recipe to cook something, but it doesn't cover the details of the ingredient, etc. I just want to look at the problem from different points of view. – 2017-01-20
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    @Rohan I want to see if there is a way to calculate the same result for P6 without using subtraction like above. Like by the product rules, instead of the sum rule. – 2017-01-20
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    @ Joffan I apologize I didn't specify my intend, I didn't mean the other methods are stupid, I think I said I was looking for more stupid hard way, which is comparatively speaking, so any method that does more step and yield the same result are considered the less sufficient ways to me, and percisely these kind of ways are what I am looking for. – 2017-01-20
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    You may consider using formulas for factoring the difference of same powers such as at the following site: http://www.themathpage.com/alg/difference-two-squares-2.htm – 2017-01-20

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There is one way I can think of on the fly.

$36^6-26^6$

$=(36^3)^2-(26^3)^2$

$=(36^3+26^3)(36^3-26^3)$

$=(36+26)(36^2+26^2-26×36)(36-26)(36^2+26^2+36×26)$

$=62×1036×10×2908$

$=1867866560$