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  • 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 man

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?

 

Option: 1

15,400


Option: 2

14,800


Option: 3

23,600

 


Option: 4

32,000


Answers (1)

best_answer

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:  \mathrm{C(5,2)=5 ! /(2 ! \times(5-2) !)=10 }

For the three consonant positions, there are 21 consonants to choose from, and we need to select 3 of them. Using the combination formula: \mathrm{ C(21,3)=21 ! /(3 ! \times(21-3) !)=1,330 .}

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: \mathrm{C(5,3)=5 ! /(3 ! \times(5-3) !)=10 . }

For the two consonant positions, there are 21 consonants to choose from, and we need to select 2 of them. Using the combination formula: \mathrm{C(21,2)=21 ! /(2 ! \times(21-2) !)=210.}

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: \mathrm{10 \times 1,330+10 \times 210=13,300+2,100=15,400 . }

Therefore, there are 15,400 different passwords possible.

Posted by

Sanket Gandhi

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