Permutation

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

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

Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

How many 3-letter words with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed?

In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?

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?

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

A coin is tossed 3 times. Find out the number of possible outcomes.