Fundamental Principle of Counting, Permutations and Combinations

How many even 4-digit whole numbers are there?

In a multiple-choice problem paper of 15 questions, the answers can be A, B, C or D. The number of different ways of answering the question paper is

How many words can be made with seven letters that start with a vowel and end with an A? Note that they do not have to be real words and letters can be repeated.

Neela has twelve different skirts, ten different tops, eight different pairs of shoes, three different necklaces and five different bracelets. How many possible ways can Neela dress?

How many five-digit numbers can be made from the digits 1 to 7 if repetition is allowed?

For her English literature course, Ruchika has to choose one novel to study from a list of ten, one poem from a list of fifteen and one short story from a list of seven. How many different choices does Ruchika have?

There are two different Geography books, five different Natural Sciences books, three different History books and four different Mathematics books on a shelf. How many different ways can they be arranged if all the books on the same subjects stand together?

A safe’s passcode is of the form PPPQQQQ, where P is any number from 0 to 9 and Q represents the letters of the alphabet. How many possible codes exist? Note that repetition of digits or letters is permitted.

Amit must choose a seven-digit PIN, and each digit can be selected from 0 to 9. How many different PINs can Amit choose?

A head boy, a head girl, two deputy head boys, and three deputy head girls must be chosen out of a student council consisting of 14 girls and 16 boys. How many ways can they be selected?

There are six equally spaced points, A, B, C, D, E and F, marked on a circle with radius R. How many convex heptagons of distinctly different areas can be drawn using these points as vertices?

There are two twin sisters among a group of 15 people. How many ways can the group be arranged in a circle to have precisely one person between the two sisters?

How many ways can 7 chocolate biscuits and 12 cheesecake biscuits be arranged to form a row of 19 biscuits?

There are 15 people in a committee. How many ways are there to group these 15 people into 3, 5, and 4?

There are six movie parts numbered from 1 to 6. Find the number of ways they can all be arranged so that Part-1 and Part-3 are never together.