How many words, with or without meaning, each of 3 vowels and 2 consonants can be formed from the letter of the word 'involute'? Can we solve it by permutations.answer: 2880?

Question

Answer

INVOLUTE => vowels : I O U E and consonants : N V L T

Number of ways of selecting 3 vowels = 4C3 = 4

Number of ways of selecting 2 consonants = 4C2 = 6

Possible number of words on arranging 3 vowels and 2 consonants = (C3)(4C2)5!

= 4 x 6 x 120

= 2880 words

Answer

Solution:
To solve this problem, we can use permutations.

Step 1: Find the number of ways to choose 3 vowels from the word 'involute'.

The word 'involute' has 4 vowels (i, o, u, e) and we need to choose 3 of them.

This can be done in 4C3 ways = 4 ways.

Step 2: Find the number of ways to choose 2 consonants from the word 'involute'.

The word 'involute' has 4 consonants (n, v, l, t) and we need to choose 2 of them.

This can be done in 4C2 ways = 6 ways.

Step 3: Find the number of ways to arrange the chosen vowels and consonants.

We have 3 vowels and 2 consonants.

These can be arranged in 5!/(3!2!) = 10 ways.

Step 4: Find the total number of words that can be formed.

The number of ways to choose the vowels and consonants and arrange them is the product of the results from steps 1, 2 and 3.

Total number of words = 4 x 6 x 10 = 240.

Step 5: Find the number of ways to arrange the letters in each word.

Each word has 5 letters, which can be arranged in 5! ways.

Total number of arrangements = 5! = 120.

Step 6: Find the total number of words with or without meaning.

The total number of words is the product of the results from steps 4 and 5.

Total number of words = 240 x 120 = 28,800.

Therefore, the number of words, with or without meaning, each of 3 vowels and 2 consonants that can be formed from the letters of the word 'involute' is 2,880.

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