Questions: How many four-letter passwords can be formed from the letters in the word "mindful" if each letter has to be distinct? There can be such passwords.

How many four-letter passwords can be formed from the letters in the word "mindful" if each letter has to be distinct?

There can be such passwords.

Transcript text: Part 1 of 2 (a) How many four-letter passwords can be formed from the letters in the word "mindful" if each letter has to be distinct? There can be $\square$ such passwords.

Solution

Solution Steps

To determine the number of four-letter passwords that can be formed from the letters in the word "mindful" with each letter being distinct, we need to calculate the number of permutations of 4 letters out of the 7 available letters.

Step 1: Determine the Number of Letters

The word "mindful" consists of 7 distinct letters: $ m, i, n, d, f, u, l $.

Step 2: Calculate the Number of Permutations

To find the number of distinct four-letter passwords that can be formed from these letters, we use the permutation formula:

\[ P(n, r) = \frac{n!}{(n - r)!} \]

where $ n = 7 $ (the total number of letters) and $ r = 4 $ (the number of letters to choose). Thus, we calculate:

\[ P(7, 4) = \frac{7!}{(7 - 4)!} = \frac{7!}{3!} = \frac{7 \times 6 \times 5 \times 4}{1} = 840 \]