Problem 27 - Algebra for College Students

Question

How many different license plates can be formed by using 3 digits followed by a single letter followed by 3 more digits? How many if the single letter can occur anywhere except last?

Step-by-Step Solution

Answer

Total combinations with a fixed letter position: 26 \times 10^6. Total combinations with the letter not in the last position: 6 \times 26 \times 10^6.

Step 1 - Determine the number of digits and letters

The license plate consists of 3 digits, followed by 1 letter, and then 3 more digits. Recall there are 10 possible digits (0-9) and 26 possible letters (A-Z).

Step 2 - Calculate combinations for the digits and letter fixed position

We need 3 digits, 1 letter, and then 3 more digits. The total number of possible combinations for the digits is \[\text{10}^3 \times \text{10}^3 = 10^6\]. The letter has \[\text{26}\] possibilities. Therefore, the total combinations for having the letter in a fixed position would be \[\text{26} \times 10^6\].

Step 3 - Place the letter in any position except the last

If the letter can be placed anywhere except the last position, it has 6 possible positions (first, second, third, fourth, fifth, or sixth). Thus, the number of ways to arrange the digits remains \[\text{10}^6\] and we have \[\text{26}\] letters and \[\text{6}\] positions for each letter. The total number of combinations is \[\text{6} \times \text{26} \times \text{10}^6\].

Key Concepts

license plate combinations

Creating unique license plates involves understanding the combinations of letters and digits that can be used. In our sample exercise, a license plate is made of 3 digits, followed by 1 letter, followed by 3 more digits. This structure dictates how many possible combinations there are.
The digits range from 0 to 9, meaning there are 10 choices available for each digit. The letter comes from the English alphabet which has 26 choices from A to Z.
To find the total possible combinations, we first calculate for the digits. If each of the 6 digits has 10 possibilities, we get \[10^3 \times 10^3 = 10^6\]. Since there's one letter in one position, we multiply by 26, leading to \[26 \times 10^6\].

permutations and combinations

In combinatorics, permutations and combinations are key concepts to determine the number of possible arrangements. While permutations pertain to different orderings of a set of items, combinations refer to selecting items without worrying about their order.
For license plates, determining how many plates can be formed given certain positions means we're dealing with a combination where each slot has a fixed set of possibilities. In our problem, the placement within those slots (the exact order) is what matters, making it somewhat similar to permutations. However, because each plate's structure (3 digits, 1 letter, 3 digits) remains fixed, calculating the total forms a straightforward combinatorial equation like \[26 \times 10^6\].
Adjusting the problem where the letter can't be at the end means adding permutations for different positions.

probability

Probability measures how likely an event is to occur. For our license plate problem, we can use probability to understand how likely it is to form a specific plate.
The probability of creating one specific combination out of all possible combinations is the inverse of the total number of combinations. From the first scenario, we calculated \[26 \times 10^6\] combinations. Therefore, the probability of any single specific combination being formed is \[\frac{1}{26 \times 10^6}\].
This calculation becomes somewhat complex if we modify our scenario. If the letter can be in 6 possible positions (not the last), then the total number of combinations becomes \[6 \times 26 \times 10^6\]. The probability for a single, specific combination in this case would thus become \[\frac{1}{6 \times 26 \times 10^6}\]. Understanding these calculations helps in grasping the vast number of different possible license plates that can exist!