Solution

Given:• We are given 8 letters- ABCDEFIO

• We need to form 4 letter words from ABCDEFIO such that it contains 2 vowels and 2 consonants

• No letter in the 4-letter word should be repeated

To find:• We need to find the number of ways in which 4-letter word can be formed such that it includes 2 vowels and 2 consonants from the 8 letters- ABCDEFIO

Approach and Working:The 8 letters- ABCDEFIO has 4 consonants and 4 vowels.

Hence, we first we will pick 2 consonants from 4 consonants and 2 vowels from 4 vowels to form the 4-letter word.

Thus,

• Total ways= ways to pick 2 consonants from 4 consonants * ways to pick 2 vowels from 4 vowels* ways to fill the four places

Ways to picks 2 consonants from 4 consonants:• 2 consonants from the 4 consonants can be picked in 4c2=6 ways

Ways to pick 2 vowels from 4 vowels:• 2 vowels from the 4 vowels can be picked in 4c2=6 ways.

Ways to fill the 4 places in the 4-letter word:• Total ways to fill the 4 places= Ways to fill the 1st place* ways to fill the 2nd place* ways to fill the 3rd place* ways to fill the 4th place.

Since there are only 4 letters, the place can at max be filled with 4 letters.

• Hence, ways to fill the first place= 4

Now, there will be 3- letters to fill the 2nd place. Hence,

• Ways to fill the 2nd place= 3

In the similar fashion,

• Ways to fill the 3rd place= 2

• Ways to fill the 1st place= 1

Hence, total ways to form the 4 letters word= 6*6*4*3*2*1= 864

Hence, the correct answer is option D.

Answer: D

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