At the end of a business conference, the ten people present all shake hands with each other once. How many handshakes will there be altogether ?

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

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

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?