Suppose that all license plates in a state have three of the letters A through Z followed by four of the digits 0 through 9. If a plate is chosen at random, what is the probability that all three letters and all four digits on the plate will be different? Round your answer to five decimal places.
A)0.018639
B)0.074556
C)0.447337
D)0.546000
E)0.080769

Question

Suppose that all license plates in a state have three of the letters A through Z followed by four of the digits 0 through 9. If a plate is chosen at random, what is the probability that all three letters and all four digits on the plate will be different? Round your answer to five decimal places. ​
A)0.018639
B)0.074556
C)0.447337
D)0.546000
E)0.080769

Quizplus · Accepted Answer

There are 26 choices for each letter and 10 choices for each digit. Therefore, there are $26^3\cdot 10^4$ possible license plates. To count the number of license plates with all three letters and all four digits different, we use the multiplication principle. For the letters, there are 26 choices for the first letter, 25 choices for the second letter (since the first letter has already been chosen), and 24 choices for the third letter (since two have already been chosen). For the digits, there are 10 choices for the first digit, 9 choices for the second digit (since the first digit has already been chosen), 8 choices for the third digit (since two have already been chosen), and 7 choices for the fourth digit (since three have already been chosen). Therefore, there are $26\cdot 25\cdot 24\cdot 10\cdot 9\cdot 8\cdot 7$ license plates with all three letters and all four digits different. The probability of choosing one of these license plates at random is:
$$\frac{26\cdot 25\cdot 24\cdot 10\cdot 9\cdot 8\cdot 7}{26^3\cdot 10^4} = 0.447337$$
Therefore, the answer is (C) 0.447337.

Suppose That All License Plates in a State Have Three