Problem 19
Question
How many different four-letter passwords can be formed from the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F},\) and \(\mathrm{G}\) if no repetition of letters is allowed?
Step-by-Step Solution
Verified Answer
The total number of different four-letter passwords that can be formed is \( 7 * 6 * 5 * 4 = 840 \)
1Step 1: Different Letter Combination Consideration
First, consider that there are 7 options for the first letter of the password, because none of the letters have been used yet and no repetition is allowed.
2Step 2: Further Letter Combination Consideration
Next, realize that once the first letter of the password has been chosen, there are now only 6 remaining unused letters for the second letter in the password. This same logic applies for the remaining two letters, where there will be 5 remaining options for the third letter, and 4 options for the last letter.
3Step 3: Calculation of Total Permutations
To find the total amount of possible four-letter passwords, multiply the number of possibilities for each letter together: 7 (possible first letters) * 6 (possible second letters) * 5 (possible third letters) * 4 (possible last letters)
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