Permutations and Combinations

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 'CORPORATION' 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?

In how many ways can the letters of the word 'LEADER' be arranged?

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?

A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the box, if at least one black ball is to be included in the draw?

In how many ways a committee, consisting of 5 men and 6 women can be formed from 8 men and 10 women?

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?