 Hello and welcome to the session. In this session we are going to discuss about combinations. In combination only a group is made and the order in which the objects are arranged is in material. Now the number of combinations of n different things taken r at a time is denoted by ncr and we have ncr is equal to n factorial upon r factorial multiplied by n minus r factorial and we have r is greater than equal to 0 and less than equal to n. Let's try and find out 10c3 this is equal to 10 factorial upon 3 factorial into 10 minus 3 that is 7 factorial this would be equal to 10 into 9 into 8 into 7 factorial upon 3 into 2 into 7 factorial. Now this 7 and this 7 gets cancelled and we have this 3 3 times is 9 and 2 5 times is 10 so this is equal to 120. Now we already know that npr is equal to n factorial upon n minus r factorial that is this is the permutation. Now we have a very important relationship between permutations and combinations which is given by npr is equal to ncr multiplied by r factorial where we have r is greater than 0 and less than equal to n. Now in this if we take r equal to n then we have mcn and this comes out to be equal to 1 then when we take r equal to 0 that is we have nc0 which says that the number of combinations of n different things taken nothing at all this is equal to 1. Then again we have a very important result which is nc n minus r is equal to ncr that is selecting r objects out of n objects is same as rejecting n minus r objects and also we have nca is equal to ncb then this implies that a is equal to b or we can say that a is equal to n minus b that is n is equal to a plus b. Another important result is ncr plus ncr minus 1 is equal to n plus 1cr suppose we have six men and five ladies and we need to find a committee of five members consisting of three men and two ladies that is this committee of five members should have three men in it and two ladies in it. Let's see in how many ways we can form this committee of five members. Now there are six men and we need three men in the committee. So the number of ways of selecting three men is equal to 6c3 and this would be equal to factorial 6 upon factorial 3 multiplied by factorial 3 and that is equal to 6 multiplied by 5 multiplied by 4 multiplied by 3 factorial upon 3 factorial multiplied by 3 into 2. Now this three factorial and this three factorial gets cancelled and we are left with 20 that is number of ways of selecting three men is 20. Now next we need to select two ladies from the given five ladies. So number of ways of selecting two ladies is given by five c2 this is equal to 5 factorial upon 2 factorial into 3 factorial which is equal to 5 into 4 upon 2 and that is equal to 10 that is number of ways of selecting two ladies is 10. So now total number of ways of selecting five members consisting of three men and two ladies is given by 6c3 multiplied by 5c2 which is equal to 20 multiplied by 10 and that is equal to 200 ways. So this completes this session hope you have understood the concept of combinations and a very important relationship between permutations and combinations.