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?

From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?

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 '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 EDUCATION be arranged so that the vowels come together?

If a cube is to be painted with red and blue colour in such a way that each face gets exactly one colour, then in how many ways this cube can be painted?