QuestionHow many unique 4-letter passwords can be formed from the 26-letter English alphabet without repeating letters? [?] ways
Studdy Solution
STEP 1
Assumptions1. The password is made up of4 letters.
. The English alphabet has26 letters.
3. Letters cannot be repeated in the password.
STEP 2
The problem can be solved using the concept of permutations. In permutations, the order of arrangement matters. Here, we are arranging4 letters out of26, and the order in which the letters are arranged matters (because "abcd" is a different password from "bcda").
The formula for permutations iswhere- is the total number of items,
- is the number of items to choose,
- is the factorial of , and- is the factorial of .
STEP 3
In this problem, is26 (the number of letters in the English alphabet) and is (the number of letters in the password). So, we can plug these values into the permutations formula
STEP 4
implify the denominator in the formula
STEP 5
The factorial of a number is the product of all positive integers less than or equal to that number. So, is the product of all positive integers from1 to26, and is the product of all positive integers from1 to22.
We can simplify as and cancel out in the numerator and denominator
STEP 6
Calculate the number of ways a password can be madeSo, there are358,800 ways to make a password using4 letters from the26-letter English alphabet, if letters cannot be repeated.
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