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!!