There is this example in my text for counting, and I understand the solution they have given. But I can't seem to find the mistake in my initial understanding which gives a different solution which I know must be wrong. I think it is important I understand where my intuition went wrong.
Q 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?
My solution was $(10 \cdot 36^5) + (10 \cdot 36^6) + (10 \cdot 36^7)$ where I thought that if I kept one slot for only digits,giving 10 choices for it,the rest of the slots could have 36 choices each.
The correct solution :
$P6 = 36^6 − 26^6 = 2,176,782,336 − 308,915,776 = 1,867,866,560$
Similarly, we have
$P7 = 36^7 − 26^7 = 78,364,164,096 − 8,031,810,176 = 70,332,353,920$
$P8 = 36^8 − 26^8 = 2,821,109,907,456 − 208,827,064,576 = 2,612,282,842,880$
$P = P6 + P7 + P8 = 2,684,483,063,360$
Please help me understand where my solution is wrong?
P.S. I found this SO question , but the answers didn't quite help understanding my fault except the one by Mark but I need more clarification but not enough rep to comment there.