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  • A password consists of 4 letters, where each letter can be either a vowel (a, e, i, o, u) or a consonant. The password must have exactly three vowels or exactly three consonants. How many diff

A password consists of 4 letters, where each letter can be either a vowel (a, e, i, o, u) or a consonant. The password must have exactly three vowels or exactly three consonants. How many different passwords are possible?
 

Option: 1

89,456


Option: 2

36,456
 


Option: 3

28,630
 


Option: 4

27,440


Answers (1)

best_answer

Case 1: Exactly three vowels:
In this case, we have to choose 3 positions for the vowels and 1 position for the consonant. There are 5 vowels to choose from and 21 consonants to choose from. The number of ways to choose 3 positions out of 4 is given by the combination formula: \mathrm{C(4,3)=4 ! /(3 ! \times(4-3) !)=4.}

The number of ways to choose 3 vowels out of 5 is given by the combination formula: \mathrm{C(5,3)=5 ! /(3 ! \times(5-3) !)=10.}

Therefore, the total number of passwords with exactly three vowels is: \mathrm{4 \times 10 \times 21=840.}
Case 2: Exactly three consonants:
In this case, we have to choose 3 positions for the consonants and 1 position for the vowel. There are 21 consonants to choose from and 5 vowels to choose from.

The number of ways to choose 3 positions out of 4 is given by the combination formula: \mathrm{C(4,3)=4 ! /(3 ! \times(4-3) !)=4.}

The number of ways to choose 3 consonants out of 21 is given by the combination formula: \mathrm{ C(21,3)=21 ! /(3 ! \times(21-3) !)=1,330.}

Therefore, the total number of passwords with exactly three consonants is: \mathrm{ 4 \times 1,330 \times 5=26,600. }

To find the total number of different passwords, we add the possibilities from both cases:

\mathrm{ 840+26,600=27,440 \text {. }}

Therefore, there are 27,440 different passwords possible.

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Gaurav

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