What is the maximum possible number of 3 letter words in English that do not contain any vowel repetition is allowed?

Text Solution

Answer : (i) 60 (ii) 125

Solution : (i) Total number of 3-letter words is equal to the number of ways of filling 3 places. First place can be filled in 5 ways by any of the given five letters. Second place can be filled in 4 ways by any of the remaining 4 letters and the third place can be filled in 3 ways by any of the remaining 3 letters. <br> So, the required number of 3-letter words `=(5xx4xx3)=60.` <br> (ii) When repetition of letters is allowed, each place can be filled by any of the 5 letters in 5 ways. <br> `therefore " the required number of ways "=(5xx5xx5)=125.`

There are four distinct letters in the word SERIES. You can either use three of the four letters or use two of the four letters by using either S or E twice.

You can use three distinct letters in $P(4, 3) = 4 \cdot 3 \cdot 2 = 24$ ways.

You can use exactly two letters if you use S or E twice. Thus, there are $C(2, 1)$ ways of choosing the repeated letter, $C(3, 2)$ ways of choosing where to place those letters in the three letter word, and $C(3, 1)$ of choosing the third letter in the word, giving

$$\binom{2}{1}\binom{3}{2}\binom{3}{1} = 2 \cdot 3 \cdot 3 = 18$$

ways to form a word with a repeated letter.

Consequently, there are $24 + 18 = 42$ distinguishable three letter words that can be formed with the letters of the word SERIES.