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?

In how many different ways can the letters of the word 'LEADING' be arranged in such a way that the vowels always come together?

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

The number of ways in which the letters of the word “ALGEBRA” can be arranged so that two A's are not adjacent, are

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?

How many words can be formed from letters of the word 'ULTRA' so that the vowels never come together?