A password consists of 5 letters, where each letter can be either a vowel (a, e, i, o, u) or a consonant. In addition, the password must have at least two vowels. How many different passwords are possible?
15,400
14,800
23,600
32,000
First, let's determine the number of passwords with exactly two vowels and three consonants:
For the two vowel positions, there are 5 vowels to choose from, and we need to select 2 of them.
This can be calculated using the combination formula:
For the three consonant positions, there are 21 consonants to choose from, and we need to select 3 of them. Using the combination formula:
Now, let's consider the number of passwords with at least three vowels:
For the three vowel positions, there are 5 vowels to choose from, and we need to select 3 of them.
Using the combination formula:
For the two consonant positions, there are 21 consonants to choose from, and we need to select 2 of them. Using the combination formula:
To find the total number of different passwords, we add the number of passwords with exactly two vowels and three consonants to the number of passwords with at least three vowels:
Therefore, there are 15,400 different passwords possible.
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