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.

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?单项选择题

类似问题

An art gallery has 8 paintings available to display, but there are only 6 spaces (in a line) to hang them. How many different arrangements are possible? There are two correct answers; select them both.

How many different licence plates can be made using two letters followed by three digits, if neither the letters nor the digits can be repeated?

In how many ways can seven friends be seated in a row at the theatre?

How many ways to arrange 4 students in a line from a group of 6?

In which case does order matter?

Identify the following as Permutations, or Combinations (no need to solve). In a race in which six automobiles are entered and there are not ties, in how many ways can the first four finishers come in?

In how many ways can seven friends be seated in a row at the theatre?

How many different licence plates can be made using two letters followed by three digits, if neither the letters nor the digits can be repeated?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单