Question: License plates: In a certain state, license plates consist of five digits from 0 to 9 followed by…

License plates: In a certain state, license plates consist of five digits from 0 to 9 followed by two letters. Assume the numbers and letters are chosen at random. Replicates are allowed.

Part: 0 / 3

Part 1 of 3

(a) How many different license plates can be formed?

The number of different license plates is \(\boxed{\phantom{0}}\).

Part: 1 / 3

Part 2 of 3

(b) How many different license plates have the letters J-O in that order?

The number of different license plates that have J-O in that order is \(\boxed{\phantom{0}}\).

Part: 2 / 3

Part 3 of 3

(c) If your name is Jo, what is the probability that your name is on your license plate? Write your answer as a fraction or a decimal, rounded to at least 5 places.

The probability that Jo is on your license plate is \(\boxed{\phantom{0}}\).

Solution

(a) To find the number of different license plates that can be formed: Each license plate consists of 5 digits followed by 2 letters. For each digit, there are 10 possibilities (0 to 9). For each letter, there are 26 possibilities (A to Z). The total number of different license plates is calculated by multiplying the number of choices for each position: \[ 10^5 \times 26^2 \] Calculating the values: \[ 10^5 = 100000 \] \[ 26^2 = 676 \] Multiply these together: \[ 100000 \times 676 = 67600000 \] So, the number of different license plates is 67,600,000. (b) To find the number of license plates with letters J-O in that order: The first five positions can be any digits, and the last two positions must be ‘J’ and ‘O’. For the 5 digits, you have: \[ 10^5 \] Calculate: \[ 10^5 = 100000 \] So, there are 100,000 different license plates that have J-O in that order. (c) To find the probability that your name “Jo” is on your license plate: We need the probability that J-O appears in the last two positions. The total number of possible license plates is 67,600,000. The number of plates with “Jo” is 100,000 (from part b). The probability is: \[ \frac{100000}{67600000} \] Simplify the fraction: \[ \frac{1}{676} \] Convert to decimal: \[ \frac{1}{676} \approx 0.001479 \] So, the probability that “Jo” is on your license plate is approximately 0.001479.