How many 4-letter words can be formed out of the letters of 'EGOIST' h

1.	How many 4-letter words can be formed out of the letters of 'EGOIST' having at least one vowel? (repetition is not allowed)1. 12 × 4!2. 15 × 4!3. 21 × 4!4. 24 × 4!
Answer» Correct Answer - Option 2 : 15 × 4! Concept: The ways of arranging n different things = n! The ways of arranging n things, having r same things and rest all are different = \(\rm n!\over r!\) The no. of ways of arranging the n arranged thing and m arranged things together = n! × m! The number of ways for selecting r from a group of n (n > r) = nCr Calculation: There are 3 vowels O, I, E Required number of ways N = 1 vowel and 3 consonants + 2 vowel and 2 consonants + 3 vowel and 1 consonants ⇒ N = [\(\rm\left(^3C_1×{^3C_3}\right) + \left({^3C_2}×{^3C_2}\right) + \left({^3C_3}×{^3C_1}\right)\)] × 4! ⇒ N = (3 + 9 + 3) × 4! ⇒ N = 15 × 4! words

How many 4-letter words can be formed out of the letters of 'EGOIST' having at least one vowel? (repetition is not allowed)1. 12 × 4!2. 15 × 4!3. 21 × 4!4. 24 × 4!

Answer» Correct Answer - Option 2 : 15 × 4!

Concept:

The ways of arranging n different things = n!
The ways of arranging n things, having r same things and rest all are different = \(\rm n!\over r!\)
The no. of ways of arranging the n arranged thing and m arranged things together = n! × m!
The number of ways for selecting r from a group of n (n > r) = nCr

Calculation:

There are 3 vowels O, I, E

Required number of ways N = 1 vowel and 3 consonants + 2 vowel and 2 consonants + 3 vowel and 1 consonants

⇒ N = [\(\rm\left(^3C_1×{^3C_3}\right) + \left({^3C_2}×{^3C_2}\right) + \left({^3C_3}×{^3C_1}\right)\)] × 4!

⇒ N = (3 + 9 + 3) × 4!

⇒ N = 15 × 4! words

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How many 4-letter words can be formed out of the letters of 'EGOIST' having at least one vowel? (repetition is not allowed)1. 12 × 4!2. 15 × 4!3. 21 × 4!4. 24 × 4!

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