 Hello and welcome to the session. In this session we will discuss how to find probability using counting rules. There are three counting rules. First is fundamental principle of counting. Second, permutation rule and third, combination rule. We are already familiar with these rules. Let us recall fundamental principle of counting. If there are E1, E2, E3 and so on up to EN events and event E1 can occur in M1 ways. Event E2 after event E1 has already occurred in M2 ways and so on. Then these events can occur in M1 into M2 into and so on up to Mn ways. We know that permutation describes the number of ways in which n distinct objects can be arranged. The number of permutations or arrangement of n distinct objects is given by n factorial. Also the number of permutations of n objects taken are at a time is denoted by pnr or can also be written as npr which is equal to n factorial upon n minus r factorial. We know that when arrangement or ordering does not matter we use combinations. It is choosing r things from a set of n objects. Thus number of combinations of n things taken are at a time will be given by ncr that is equal to n factorial upon n minus r factorial into r factorial. Now we will use these rules in finding probabilities. Now we shall discuss how to find probability using counting rules. Let us consider the following example. If two cards are drawn simultaneously from a well shuffled deck of 52 cards then what is the probability that both cards drawn are kings? Let us start with its solution. We know that probability of an event is given by favorable outcomes upon total number of outcomes. Since we have to draw two cards simultaneously from 52 cards so the total number of outcomes will be 52 c2. Here we want to choose only two kings as there are four kings in the deck of 52 cards. So favorable outcomes will be 4 c2 so probability of drawing two kings will be given by favorable outcomes upon total number of outcomes and that is equal to 4 c2 upon 52 c2. Now using this formula here we write 4 factorial upon 4 minus 2 factorial into 2 factorial whole upon 52 factorial upon 52 minus 2 factorial into 2 factorial and this is equal to 4 factorial upon 2 factorial into 2 factorial whole upon 52 factorial upon 50 factorial into 2 factorial. Now we are going to solve it. Here 4 factorial upon 2 factorial into 2 factorial into 50 factorial into 2 factorial upon 52 factorial. Now this can be written as 4 into 3 into 2 factorial upon 2 factorial into 2 factorial and 2 factorial is 2 into 1 so we have 2 into 50 factorial into 2 factorial that is 2 upon 52 factorial can be written as 52 into 51 into 50 factorial. Here 2 factorial cancels with 2 factorial 2 into 1 is 2 and 2 into 2 is 4. Now here 50 factorial cancels with 50 factorial and 2 into 1 is 2. 2 into 26 is 52. Again 2 into 1 is 2 and 2 into 13 is 26. Also 3 into 1 is 3 and 3 into 17 is 51. So this is equal to 1 upon 13 into 17 that is equal to 1 upon 221. So probability of drawing to a king's is 1 upon 221. So here we have seen that we can find probabilities by using combinations. Let us consider one more example. A 3 digit number is formed using digits 1 to 9 where repetition of digits is allowed. Find the probability that the number formed ends with digit 2. Let us start with its solution. Here we have to form a 3 digit number using digits 1 to 9. So there are total 9 numbers. So let us first find the number of ways to form a 3 digit number using these 9 numbers. Now in this 3 digit number unit's place can be filled in 9 ways as repetition is allowed. So 10's place can also be filled in 9 ways and 100's place can also be filled in 9 ways. So by fundamental principle of counting the total number of ways of forming the 3 digit number is 9 into 9 into 9 that is equal to 729. So total number of outcomes will be equal to 729. Now we want to find the number of ways in which a 3 digit number can be formed such that it ends in digit 2. So now the unit's place is fixed by number 2. So the number of ways in filling unit's place is 1. Now number of ways for filling 10's place is 9 and number of ways for filling 100's place is also 9. So by fundamental principle of counting the total number of ways of forming the 3 digit number ending with digit 2 will be equal to 9 into 9 into 1 that is equal to 81. So favorable outcomes will be equal to 81. Now here total number of outcomes are given by 729 and favorable outcomes are given by 81. So probability that the number formed ends with digit 2 is equal to favorable outcomes upon total number of outcomes and this will be equal to 81 upon 729 that is equal to 1 by 9. Thus we can use fundamental principle of counting to find probabilities. This completes our session. Hope you enjoyed this session.