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# Permutations & Combinations

1.

In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?

The word 'OPTICAL' contains 7 different letters.

When the vowels OIA are always together, they can be supposed to form one letter.

Then, we have to arrange the letters PTCL (OIA).

Now, 5 letters can be arranged in 5! = 120 ways.

The vowels (OIA) can be arranged among themselves in 3! = 6 ways.

Required number of ways = (120 x 6) = 720.

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2.

In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?

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3.

How many 4-letter words with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed?

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4.

In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?

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5.

In how many different ways can the letters of the word 'DETAIL' be arranged in such a way that the vowels occupy only the odd positions?