 Hi, and how are you all today? I'm Priyanka. The question says in an examination a question paper consists of 12 questions divided into two parts that is part one and part two containing five and seven questions respectively. A student is required to attempt eight questions in all selecting at least three from each part. In how many ways can a student select the questions? Now for this we should be well versed with a formula for number of combinations that is ncr is equal to n factorial divided by r factorial n minus r factorial. The knowledge of this formula is the key idea we are going to use to proceed on with the solution. Now we are given question paper is having two parts that is part one and part two. Part one is containing five questions and part two is containing seven questions. Now eight questions now eight questions can be chosen in the following three ways. Questions from part one and five questions from part two or we can choose four questions from part one and similarly four questions from part two or we can choose five questions from part one or three questions from part two, right? These are the three ways in which we can choose eight questions. Now the required number of ways for selecting three questions from part one will be out of five we can select three questions whereas five questions from part two we have seven questions in part two and we can select five. So this is our required combination for three questions from part one and five questions from part two. Further for this b part out of five questions we can choose four from part one or seven out of seven questions we are taking four questions from part four. So there are five c by four multiplied by seven c four ways of choosing these eight questions. Similarly over here we have five c five multiplied by seven c three, right? Now we will be solving them and adding the products to each other that is by applying the multiplication principle we got that and the total number of ways will be obtained by their sum. So it is equal to total number of ways of choosing eight questions from part one and two together. On simplifying we have five c three can be written as five factorial divided by three factorial five minus three factorial multiplied by seven factorial divided by five factorial seven minus five factorial plus five factorial divided by four factorial five minus four will be written as one factorial seven factorial four factorial three factorial added to five factorial by five factorial zero factorial seven factorial by three factorial four factorial this will be one further we have five factorial divided by three factorial two factorial multiplied by seven factorial five factorial multiplied by two factorial let us simplify it here we are left with seven factorial which can be written as seven six five four three factorial divided by three factorial two factorial can be written as two multiplied by one plus five factorial can be written as five multiplied by four factorial divided by four factorial multiplied by seven into six into five into four factorial divided by four factorial into three factorial plus seven six five into four factorial divided by three two one divided by 3, 2, 1 multiplied by 4 factorial. Further we have 3 factorial here can be written as 3 multiplied by 2 multiplied by 1 and on simplifying we are left with 7 multiplied by 6 multiplied by 5 plus 5 multiplied by 7 multiplied by 5 plus 5 multiplied by 7 and the product it can further be written as 210 plus 175 plus 35 which is equal to 420. So this is our required answer to the part. I hope you enjoyed this session. Use the formula for counting of combinations and take care.