 Hello and welcome to the session. In this session we will discuss the use of permutations and combinations to find probabilities of compound events. Let us first look at the following definitions. Now permutation is an arrangement or listing in which order is important. The number of permutations of n things taken R at a time is given by NPR which is equal to n factorial upon n minus R to the factorial second most combination. Now combination is an arrangement or listing in which order is not important. In probabilities these are used when things are chosen without replacement. Now the number of combinations of n things taken R at a time is given by NCR which is equal to n factorial upon R factorial into n minus R to the factorial. Also you must note that n factorial is equal to n into n minus 1 the whole into n minus 2 the whole and so on up to 3 into 2 into 1. Now let us discuss fundamental quantum principle. Now if event A possible outcomes event B has n possible outcomes when event A followed by event B has n into n possible outcomes. Now we should make use of these results in finding probabilities of compound events. Now let us consider an example. Now in this example it is given that a committee of five members is to be constituted. From a group of six males and eight females if the selection is made randomly find the probability that committee has three female members and two male members that committee has all male members. Now let us start with its solution. Now we know that probability of an event is equal to number of favorable outcomes upon total number of outcomes. Now total number of candidates is equal to six males plus eight females which is equal to fourteen candidates. Since fourteen candidates we have to select five candidates which means out of fourteen five that is number of combinations fourteen at a time which is that is fourteen C five is equal to fourteen C five. Now we have to find favorable outcomes for the event of selecting three female members and randomly select three females from followed by selecting any two males from it means out of eight females choose three females followed by out of six males choose two males that is eight C three followed by six C two so fundamental principle of counting number of favorable outcomes is equal to eight C three into six C two so the required probability P of selecting three females is equal to number of favorable upon total number of outcomes so this is equal to C three into six C two whole upon fourteen C five. Now we know that NCR is equal to N factorial over R factorial into N minus R whole factorial so here this is equal to now eight C three will be eight factorial upon three factorial into H minus three whole factorial into six C two will be six factorial upon two factorial into six minus two whole factorial whole upon now fourteen C five will be fourteen factorial upon five factorial into fourteen minus five whole factorial and this is equal to eight factorial upon three factorial into five factorial into six factorial upon two factorial into four factorial point upon fourteen factorial upon five factorial into now fourteen minus five is nine so it will be nine factorial now on calculating eight factorial upon three factorial into five factorial is equal to eight into seven and this is equal to fifty six and six factorial upon two factorial into four factorial is equal to fifteen and fourteen factorial upon five factorial into nine factorial is equal to two thousand and two putting all these values here this is equal to fifty six into fifteen whole upon two thousand two now two into twenty eight is fifty six and two into one thousand one is two thousand two then seven into four is twenty eight and seven into one forty three is one thousand and one so this is equal to four into fifteen that is sixty upon one forty three so probability of selecting three females and two males is sixty upon one forty three now here although does not matter we could have selected male members first and female members later then also probability will remain same thus whenever we have to choose our things from given and things we make use of combinations now in the second part we want to find probability of selecting all males now total number of outcomes is equal to fourteen three five which is equal to two thousand two now where we have to select all males so favorable outcomes means no female is selected and all males are selected that is zero female males as here we have to select five candidates is equal to c zero into six from eight female members we have to choose zero female member and from six male members we have to choose five male members and this is equal to eight factorial upon zero factorial into eight minus zero whole factorial into six factorial upon five factorial into six minus five whole factorial now you must know that zero factorial is equal to one so this is equal to eight factorial upon one into eight minus zero factorial that is eight factorial into six into five factorial upon five factorial into one factorial now this is equal to one into now one factorial is one so this is one into six which is equal to six so probability P of selecting all males is equal to number of favorable outcomes that is six upon total number of outcomes that is two thousand two so this is equal to six upon two thousand two which is equal to three upon one thousand one so probability of selecting all males is equal to three upon one thousand one so in this session we have discussed the use of combinations and combinations to find probabilities of compound events and this completes our session hope you all have enjoyed the session.