Question 12 and 13, Exercise 6.2

Solutions of Question 12 and 13 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Find the numbers of arrangement of letters of the word VOWEL in which vowels may occupy old places?

Solution.

Total letters $=5$
Total possible arrangements of these $5$ letters $=5!=120$
Vowel letters in the given word are ' $O$ ' and ' $E$ '.
Out of five, second and fourth places are even and
possible arrangements of two vowels on two places is $21=2$
and possible arrangements of $3$ consonants an odd placés is $3!= 6$
Total arrangements when $O$ \& $E$ occupy even places $=6 \times 2=12$
Total arrangements when $0$ \& $E$ occupy odd places $=120-12=108$

In howmany ways can of letters of the word MACHINE be arranged so that all the vowels are never togather?

Solution.

Total letters in word $=7$
Possible arrangements of $7$ letters $=71=5040$
There are $3$ vowels $A, I$ and $E$ in word.
To find the arrangements when all vowels are together,
we treat all $3$ vowels as single element and
calculate permutations of this element along with remaining $4$ consonant letters i.e.,
$$5!=120$$
arrangements of $3$ vowels $=6$
so total $$6 \times 120=\mathbf{7 2 0}$$
Hence out of $5040$ arrangements in $720$ arrangements $3$ vowels are together.
So arrangements when vowels are not together is $5040-720=4320$.

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