[Math] How many valid passwords are there if it must be 6 characters long, and contain at least 2 lowercase letters

combinatorics

Question

Students are required to create 6-character long passwords to access the library. The letters must be from lowercase letters or digits. Each password must contain at least 2 lowercase letters. How many valid passwords are there?

I would like to know if my steps to solving this question, and the final answer, are correct.

Steps:

  1. All passwords = 36^6

  2. No letters = 10^6

  3. 1 letter = 10^5 + 26^1

  4. Subtract 2,3 from 1

–END–

Best Answer

The subtraction approach is right, but each individual calculation needs to be corrected

$$\text{Total Number of Passwords} = 36^6$$ This is fine

$$\text {Number of passwords with no letters} = 10^6$$

This is also alright

$$\text{Number of passwords with exactly one letter} = {6\choose 1}.{26 \choose 1}.10^5$$

Here, we first choose the place where the alphabet will come, then we have to choose a letter from the available 26, and the remaining places have to be filled with numbers.

I hope this clears up any doubts

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