How many words with or without meaning can be formed using all the letters of the word LAUGHTER if (i) the wor

Exercise 6.3 · Q10

Q. How many words with or without meaning can be formed using all the letters of the word LAUGHTER if

(i) the words start with L but does not end with R

(ii) no two vowels come together

(iii) the relative positions of vowels and consonants remains unchanged

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LAUGHTER $=$ L,A,U,G,H,T,E,R — 8 distinct letters (vowels A,U,E — 3; consonants L,G,H,T,R — 5).

^nP_r=\dfrac{n!}{(n-r)!}

(i) Start with L but does not end with R

Words starting with L: fix L in position 1; remaining 7 letters fill the rest: $7!=5040$ .
Words starting with L and ending with R: fix L first, R last; remaining 6 letters fill the middle: $6!=720$ .
Words starting with L but NOT ending with R $=5040-720=4320$ (complementary counting).

(ii) No two vowels come together

Arrange the 5 consonants (L,G,H,T,R) first: $5!=120$ ways. This creates 6 gaps (before, between, after).
Place the 3 distinct vowels (A,U,E) into 3 of these 6 gaps (one per gap, order matters): $^6P_3=6\times5\times4=120$ .
Total $=120\times120=14400$ .

(iii) Relative positions of vowels and consonants unchanged …

Q.How many words with or without meaning can be formed using all the letters of the word LAUGHTER if