How many words each of 3 vowels and 2 consonant can be formed from the letters of the word in Bollywood?

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The word is 'INVOLUTE'
               Number of consonants = 4
                     Number of vowels = 4.
The words formed should contain 3 vowels and 2 consonants.
The problems becomes:
(i)                 Select 3 vowels out of 4.

                   Number of selections =
(ii)         Select two consonants out of 4.
                    Number of selections = 
(iii)  Arrange the five letters (3 vowels + 2 consonants) to form words.
                        Number of permutations = 5!
(iv)  Apply fundamental principle of counting:

                 Number of words formed = 

                                                  = 

                                                  = 4 x 6 x 120 = 2880  
Hence, the number of words formed  = 2880

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Given,

The word is INVOLUTE. 

We have 4 vowels namely I,O,U,E, and consonants namely N,V,L,T. 

We need to find the no. of words that can be formed using 3 vowels and 2 consonants which were chosen from the letters of involute. 

Let us find the no. of ways of choosing 3 vowels and 2 consonants and assume it to be N1. 

⇒ N1 = (No. of ways of choosing 3 vowels from 4 vowels) × (No. of ways of choosing 2 consonants from 4 consonants) 

⇒ N1 = (4C3) × (4C2) 

We know that,

nCr = \(\frac{n!}{(n-r)!r!}\)

And also,

n! = (n)(n – 1)......2.1

⇒ N1 = 4 × 6 

⇒ N1 = 24

Now,

We need to find the no. of words that can be formed by 3 vowels and 2 consonants. 

Now,

We need to arrange the chosen 5 letters. 

Since every letter differs from other. 

The arrangement is similar to that of arranging n people in n places which are n! ways to arrange. So, the total no. of words that can be formed is 5!. 

Let us the total no. of words formed be N. 

⇒ N = N1 × 5! 

⇒ N = 24 × 120 

⇒ N = 2880 

∴ The no. of words that can be formed containing 3 vowels and 2 consonants chosen from INVOLUTE is 2880.

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