Basic rules of counting are that you multiply when independent events or scenarios are being considered together and you sum them up if they are being considered separately from each other. Here is an account of some specialized rules of counting which can come handy:
Rule No. 1:
A coin has two possible outcomes: head and tail. If you like to flip it 3 times then total number of sequences you can have is 23 = 8. Similarly, for a die which has six possible outcomes and you like to roll it 2 times then number of possible sequences you can have is 62= 36
For k possible outcomes of an operation which you like to repeat n times, total number of sequences is equal to kn
Rule No. 2:
Total number of sequences of flipping a coin ‘and’ rolling die will be simple 2 * 6 = 12. Similarly, total number of sequences of flipping a coin 3 times ‘and’ rolling a die 2 times will be 23 * 62 = 288
For k1, k2,…., ki different types of possible outcomes each of which you like to repeat n1, n2,…., nj times respectively, total number of sequences is equal to k1n1 * k2n2 *….* kinj
Rule No. 3:
In a library, you have 3 empty slots for 3 books. Number of ways you can fill up the first space is 3. Let’s suppose you picked a book out of those 3 books and placed it in an empty slot on the shelf. Now, in order to fill up second empty slot you are left with 3 – 1 = 2 books. Why? Because you one of the 3 books is already there on shelf. Similarly, total number of way to fill up third place will be 3 – 2 = 1 book. Now, total number of sequence you can have for arranging the books will be 3 * 2 * 1 = 6.
Total number of sequences you can have by ordering n objects is n * (n -1) * (n – 2) * …. * 1.
There is a special term for such type of counting i.e. Permutations.