A company codes its customers by giving each customer an eight character code. The first 3 characters are the letters A, B and C in any order and the remaining 5 are the digits 1,... A company codes its customers by giving each customer an eight character code. The first 3 characters are the letters A, B and C in any order and the remaining 5 are the digits 1, 2, 3, 4 and 5 also in any order. If each letter and digit can appear only once, what is the number of customers the company can code?
Understand the Problem
The question is asking us to determine the total number of unique customer codes that can be generated using a specific rule for characters. The first part consists of selecting and arranging 3 letters from A, B, and C, and the second part consists of selecting and arranging 5 digits from 1, 2, 3, 4, and 5. We will solve this by calculating the permutations of letters and digits separately and then multiplying the results.
Answer
$720$
Answer for screen readers
The total number of unique customer codes that can be generated is $720$.
Steps to Solve
- Calculate permutations of letters
We need to select and arrange 3 letters from the set {A, B, C}.
Since we are choosing 3 out of 3 letters, the number of arrangements (permutations) can be calculated using the formula for permutations:
$$ P(n, r) = \frac{n!}{(n - r)!} $$
Here, ( n = 3 ) (the total number of letters) and ( r = 3 ) (the letters we’re arranging). Thus:
$$ P(3, 3) = \frac{3!}{(3 - 3)!} = \frac{3!}{0!} = 3! = 6 $$
- Calculate permutations of digits
Next, we need to select and arrange 5 digits from the set {1, 2, 3, 4, 5}.
Since we are also using all 5 digits, we will use the same permutation formula:
$$ P(n, r) = \frac{n!}{(n - r)!} $$
Here, ( n = 5 ) (the total number of digits) and ( r = 5 ). Thus:
$$ P(5, 5) = \frac{5!}{(5 - 5)!} = \frac{5!}{0!} = 5! = 120 $$
- Total unique customer codes
To find the total number of unique customer codes, we multiply the numbers of permutations of letters and digits:
$$ \text{Total Codes} = P(3, 3) \times P(5, 5) = 6 \times 120 = 720 $$
The total number of unique customer codes that can be generated is $720$.
More Information
The calculation shows that by selecting and arranging characters independently for letters and digits, we can generate a large diversity of unique codes. This approach can be applied to other similar problems involving permutations.
Tips
- Confusing combinations with permutations. Remember that permutations consider the arrangement, while combinations do not.
- Miscalculating factorials when determining the number of arrangements.
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