A password for a computer system requires exactly 6 characters. Each character can be either one of the 26 letters from A to Z or one of the ten digits from 0 to 9. The first character must be a letter and the last character must be a digit. How many different possible passwords are there?
(a) Less than $ {{10}^{7}} $
(b) Between $ {{10}^{7}} $ and $ {{10}^{8}} $
(c) Between $ {{10}^{8}} $ and $ {{10}^{9}} $
(d) Between $ {{10}^{9}} $ and $ {{10}^{10}} $

Hint: It is given that the first character of the password must be a letter, so the first character of the password has 26 options, i.e., one of the letters from A to Z. Also, it is given that the last character of the password must be a digit, so, we have 10 digits to fill the last character of the password. Rest of the 4 characters of the password can be any letter or digit, i.e., a total of 36 options for each of the 4 places in the password. Multiply the options for each case to get the answer.

Complete step-by-step answer:
It is given that the first character of the password must be a letter, so the first character of the password has 26 options, i.e., one of the letters from A to Z. Also, it is given that the last character of the password must be a digit, so, we have 10 digits, i.e., 0 to 9 to fill the last character of the password. Rest of the 4 characters of the password can be any letter or digit, i.e., a total of 36 options, 10 digits and 26 letters for each of the 4 places in the password. As we have to multiply all the places simultaneously to make a password, we will multiply the possibilities of each place to get the total number of possible passwords.
 $ \text{Total number of passwords possible}=26\times 36\times 36\times 36\times 36\times 10=436700160 $
As there are 9 digits in the answer and is not equal to $ {{10}^{8}} $ , we can say that 436700160 is greater than $ {{10}^{8}} $ but is surely less than $ {{10}^{9}} $ .
Hence, the answer to the above question is option (c).

Note: There is no other way to solve the question without multiplying the terms. Also, the multiplication might look difficult, but all are integers, so it is not that tough as well. The other thing to keep in mind is that you don’t miss the constraints given in the question that the first character must be a letter and the last character must be a digit.