How many 6 character passwords can be made from the 12 characters 1; 2; 3; 4; 5; 6; A;B;C;D;E; F if: (a) Adjacent characters can’t be the same. (b) The password must contain an equal number of letters and digits (in any order and repeats allowed).
The Correct Answer and Explanation is:
To find how many different passwords are possible with the given rules, we must calculate the total number of combinations:
- The password has 6 characters total.
- The first 4 characters are numbers.
- The last 2 characters are letters.
Step 1: Counting number options
Each number can be any digit from 0 to 9. This means there are 10 choices for each of the 4 number positions:10×10×10×10=104=10,00010 \times 10 \times 10 \times 10 = 10^4 = 10{,}00010×10×10×10=104=10,000
Step 2: Counting letter options
Each letter can be any uppercase letter from A to Z. There are 26 letters in the English alphabet. So, for 2 letter positions:26×26=67626 \times 26 = 67626×26=676
Step 3: Multiply the two results
Now multiply the number combinations by the letter combinations:10,000×676=6,760,00010{,}000 \times 676 = 6{,}760{,}00010,000×676=6,760,000
✅ Final Answer:
6,760,000 different passwords are possible.
Explanation
This password problem involves basic counting principles in combinatorics. When choosing items in sequence where each choice is independent of the others, we multiply the number of choices at each step.
We are told that a password must be made up of 6 characters: the first four must be digits and the last two must be letters.
There are 10 digits (0 through 9), and for each of the four number positions in the password, any of these 10 digits can be used. This gives us:10×10×10×10=104=10,00010 \times 10 \times 10 \times 10 = 10^4 = 10{,}00010×10×10×10=104=10,000
This means there are ten thousand ways to arrange the numeric portion.
Next, for the two letters at the end, each position can be filled with any of the 26 uppercase letters (A through Z). Since each letter can be repeated, we again multiply:26×26=67626 \times 26 = 67626×26=676
Finally, we combine the two parts: the numeric portion and the letter portion. Since they are chosen independently, we multiply the results:10,000×676=6,760,00010{,}000 \times 676 = 6{,}760{,}00010,000×676=6,760,000
Therefore, there are 6,760,000 possible unique passwords that fit this structure. This kind of problem is common in computer security and helps in understanding how secure a password format might be.
