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Chapter 6: Problem 30

How many license plates can be made using either three uppercase English letters followed by three digits or four uppercase English letters followed by two digits?

Short Answer

Expert verified
63,273,600 license plates can be made.

Step by step solution

01

Determine the Number of Combinations for 3 Letters and 3 Digits

First, calculate the number of possible license plates made up of three uppercase English letters followed by three digits. There are 26 English letters and 10 digits.
02

Compute Combinations for Each Section

For the first section (3 letters), the combinations are: \[26^3\] . For the second section (3 digits), the combinations are: \[10^3\] .
03

Calculate Total Combinations for First Case

Multiply the number of combinations of letters by the number of combinations of digits: \[(26^3) \times (10^3)\].
04

Determine Number of Combinations for 4 Letters and 2 Digits

Next, calculate the number of possible license plates made up of four uppercase English letters followed by two digits.
05

Compute Combinations for Each Section

For the first section (4 letters), the combinations are: \[26^4\] . For the second section (2 digits), the combinations are: \[10^2\] .
06

Calculate Total Combinations for Second Case

Multiply the number of combinations of letters by the number of combinations of digits: \[(26^4) \times (10^2)\].
07

Add Both Cases Together

Sum the total number of combinations from the first case and the second case: \[(26^3 \times 10^3) + (26^4 \times 10^2)\] .
08

Calculate the Final Result

Perform the actual calculations: \[26^3 = 17,576\] \[10^3 = 1,000\] \[26^4 = 456,976\] \[10^2 = 100\] \[(17,576 \times 1,000) + (456,976 \times 100) = 17,576,000 + 45,697,600 = 63,273,600\] }

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinations
In combinatorics, combinations are a way to calculate the number of ways a certain number of items can be selected from a larger pool, where the order does not matter. For license plates, when considering combinations of letters and digits without repetition, we calculate all possible outcomes for each independent set.
Permutations
Permutations refer to the arrangement of items where the order does matter. In the context of license plates, each unique arrangement of letters and digits counts as a different permutation. That鈥檚 why we need to consider permutations when calculating the number of possible license plates.
Probability Calculation
Probability calculation is about determining the likelihood of a specific outcome. In license plate problems, once we've computed the total number of possible permutations, we can calculate the probability of randomly generating a specific license plate by dividing the number of favorable outcomes by the total number of possible outcomes.
Exponential Growth
Exponential growth occurs when quantities increase by multiplying at a constant rate. License plate combinations grow exponentially with each added letter or digit鈥攅vident in our calculations. For instance, moving from three to four letters increases the possible combinations significantly due to the exponential nature of the calculation \(26^n\).

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Most popular questions from this chapter

Place these permutations of \(\\{1,2,3,4,5\\}\) in lexicographic order: \(43521,15432,45321,23451,23514,\) \(14532,21345,45213,31452,31542\)

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