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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 also contain at least one vowel and one conson

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 also contain at least one vowel and one consonant. How many different passwords are possible?

 

Option: 1

32,624 ways


Option: 2

28,000 ways


Option: 3

50,185 ways


Option: 4

11,024 ways


Answers (1)

best_answer

To solve this, we will consider the different cases based on the composition of the password:

Case 1: One vowel and three consonants:

Choose 1 vowel from the 5 available vowels (5 choices).

Choose 3 consonants from the 21 available consonants (21 choices for the first consonant, 20 choices for the second consonant, and 19 choices for the third consonant).

The total number of ways for this case is: 5 choices × 21 choices × 20 choices × 19 choices = 39,900 ways.

Case 2: Two vowels and two consonants:

Choose 2 vowels from the 5 available vowels (5 choices for the first vowel, 4 choices for the second vowel).

Choose 2 consonants from the 21 available consonants (21 choices for the first consonant, 20 choices for the second consonant).

The total number of ways for this case is: 5 choices × 4 choices × 21 choices × 20 choices = 8,400 ways.

Case 3: Three vowels and one consonant:

Choose 3 vowels from the 5 available vowels (5 choices for the first vowel, 4 choices for the second vowel, 3 choices for the third vowel).

Choose 1 consonant from the 21 available consonants (21 choices).

The total number of ways for this case is: 5 choices × 4 choices × 3 choices × 21 choices = 1,260 ways.

Case 4: Four vowels and no consonants:

Choose 4 vowels from the 5 available vowels (5 choices for each vowel).

The total number of ways for this case is: 5 choices × 5 choices × 5 choices × 5 choices = 625 ways.

To find the total number of possible passwords, we sum the possibilities for each case:

Total number of ways = 39,900 ways + 8,400 ways + 1,260 ways + 625 ways = 50,185 ways.

 

Posted by

manish painkra

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