The number of all possible three-letter words | vedclass

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Question

(C) The word $FEBRUARY$ consists of $8$ letters: $\{F, E, B, R, U, A, R, Y\}$. The distinct letters are $\{F, E, B, R, U, A, Y\}$.
There are $3$ vowe

Vedclass · Accepted Answer

$120$

Vedclass · Answer

(C) The word $FEBRUARY$ consists of $8$ letters: $\{F, E, B, R, U, A, R, Y\}$. The distinct letters are $\{F, E, B, R, U, A, Y\}$.
There are $3$ vowels: $\{E, U, A\}$ and $5$ consonants: $\{F, B, R, R, Y\}$.
We need to form a $3$-letter word with a vowel in the middle position.
Case $1$: The letter $R$ is not used.
We have $7$ distinct letters: $\{F, E, B, U, A, Y\}$.
- Middle position: Choose $1$ vowel out of $3$ in $^3C_1 = 3$ ways.
- First and third positions: Choose $2$ distinct letters from the remaining $6$ letters in $^6P_2 = 6 	imes 5 = 30$ ways.
- Total for Case $1$: $3 	imes 30 = 90$ ways.
Case $2$: The letter $R$ is used.
Since $R$ is a consonant,it must occupy either the first or the third position.
- Middle position: Choose $1$ vowel out of $3$ in $3$ ways.
- Position of $R$: Choose $1$ position out of $2$ (first or third) in $2$ ways.
- Remaining position: Choose $1$ letter from the remaining $5$ letters (excluding the chosen vowel and $R$) in $5$ ways.
- Total for Case $2$: $3 	imes 2 	imes 5 = 30$ ways.
Total number of words = $90 + 30 = 120$.

The number of all possible three-letter words that can be formed by choosing three letters from the letters of the word $FEBRUARY$ so that a vowel always occupies the middle place is:

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