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

How many words can be formed from the letters of the word 'ENGINEER'?

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

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

How many words, no matter if they are meaningless, can be formed by the letter, of the word ‘DIARY’?

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?