Question

1. a smartphone lets you set a 4-digit passcode using the digits 0–9. digits can repeat. how many possible passcodes exist?

Explanation:

Step1: Identify the number of choices for each digit

For each of the 4 digits in the passcode, we can use any digit from 0 - 9. So, there are 10 choices (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) for each digit.

Step2: Apply the multiplication principle

The multiplication principle of counting states that if there are \(n_1\) ways to do the first task, \(n_2\) ways to do the second task, \(\dots\), \(n_k\) ways to do the \(k\)-th task, then the total number of ways to do all \(k\) tasks together is \(n_1\times n_2\times\cdots\times n_k\).

Here, for a 4 - digit passcode, the number of choices for the first digit \(n_1 = 10\), the number of choices for the second digit \(n_2=10\), the number of choices for the third digit \(n_3 = 10\) and the number of choices for the fourth digit \(n_4=10\).

So the total number of possible passcodes is \(10\times10\times10\times10=10^{4} = 10000\).

Answer:

10000