Question

question 4 of 6
many banks require customers who use the automated teller machine (atm) to enter a four - digit password before they begin a transaction.
(a) how many possible four - digit passwords are there?
(b) how many four - digit passwords contain no 3s?

Explanation:

Step1: Analyze part (a)

Each digit of the four - digit password can be chosen from 0 - 9. So, for each of the four positions, there are 10 choices.
By the multiplication principle, the total number of four - digit passwords is \(10\times10\times10\times10\).
$$10\times10\times10\times10 = 10^{4}=10000$$

Step2: Analyze part (b)

If no digit is 3, then each digit of the four - digit password can be chosen from 9 non - 3 digits (0 - 2, 4 - 9).
By the multiplication principle, the number of four - digit passwords with no 3s is \(9\times9\times9\times9\).
$$9\times9\times9\times9=9^{4}=6561$$

Answer:

(a) \(10000\)
(b) \(6561\)