Questions: The access code for a car's security system consists of four digits. Each digit can be 0 through 9 How many access codes are possible of: a. Each digit can be repeated. b. Each digit can be used only once and not repeated. c. Each digit can be repeated, but the first digit cannot be 0 or 1.

The access code for a car's security system consists of four digits. Each digit can be 0 through 9
How many access codes are possible of:
a. Each digit can be repeated.
b. Each digit can be used only once and not repeated.
c. Each digit can be repeated, but the first digit cannot be 0 or 1.

Transcript text: The access code for a car's security system consists of four digits. Each digit can be 0 through 9 How many access codes are possible of: a. Each digit can be repeated. b. Each digit can be used only once and not repeated. c. Each digit can be repeated, but the first digit cannot be 0 or 1.

Solution

Solution Steps

Solution Approach

a. Since each digit can be repeated and there are 10 possible digits (0-9) for each of the four positions, the total number of access codes is \(10^4\).

b. If each digit can be used only once, the first digit has 10 options, the second has 9 (since one digit is used), the third has 8, and the fourth has 7. Multiply these together to get the total number of access codes.

c. If each digit can be repeated but the first digit cannot be 0 or 1, the first digit has 8 options (2-9), and the remaining three digits each have 10 options. Multiply these together to get the total number of access codes.

Step 1: Calculate Total Codes with Repetition

For part (a), where each digit can be repeated, the total number of access codes is calculated as: \[ 10^4 = 10000 \]

Step 2: Calculate Total Codes without Repetition

For part (b), where each digit can be used only once, the total number of access codes is calculated as: \[ 10 \times 9 \times 8 \times 7 = 5040 \]

Step 3: Calculate Total Codes with Restricted First Digit

For part (c), where each digit can be repeated but the first digit cannot be 0 or 1, the total number of access codes is calculated as: \[ 8 \times 10 \times 10 \times 10 = 8000 \]