If a 4-character password can include 26 lowercase letters, 26 uppercase letters, 10 digits, and 8 special characters, how many possible combinations exist if each character can only be used once?

Question

BlackTom AI · Accepted Answer

Start by identifying the total pool of available characters. There are 26 lowercase letters + 26 uppercase letters + 10 digits + 8 special characters, which sums to 70 distinct characters.
Since each character can only be used once in the 4-character password, the number of possible passwords is the number of 4-permutations of 70, written as P(70,4) = 70 × 69 × 68 × 67.
Now compute step by step:
- 70 × 69 = 4,830
- 4,830 × 68 = 328,440
- 328,440 × 67 = 22,005,480

Option-by-option analysis:
- Option 1: 22005480. This matches the calculated value 22,005,480 exactly, so it is consistent with P(70,4).
- Option 2: 916,895. This is far smaller than the correct permutation count and cannot be the result of selecting 4 distinct characters in order from 70.
- Option 3: 21,686,560. While closer in magnitude, this does not equal 70 × 69 × 68 × 67, indicating a miscalculation, perhaps missing a multiplication factor or an incorrect assumption about order or uniqueness.
- Option 4: 22,324,400. This is also close to the correct scale but does not equal the exact product 70 × 69 × 68 × 67, suggesting a misstep in one of the multiplications.

Understanding the core concept helps here: counting passwords of fixed length with no repeated characters requires ordered selections without repetition, which is precisely the permutation count P(n, k). In this scenario, n = 70 and k = 4, so the exact count is 70 × 69 × 68 × 67, yielding 22,005,480. The first option exactly matches this result, while the others deviate due to arithmetic or conceptual errors related to order or repetition assumptions.

MTH-1235-001 - 51223 - 202505 Final Exam

If a 4-character password can include 26 lowercase letters, 26 uppercase letters, 10 digits, and 8 special characters, how many possible combinations exist if each character can only be used once?

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