 Hi and how are you all? I am Priyanka. The question says a committee of seven has to be formed from nine boys and four girls. In how many ways can this be done when the committee consisted of exactly three girls, at least three girls, at most three girls? So here we will be using the formula for number of combination that is nCr is equal to n factorial divided by r factorial multiplied by n minus r factorial. And this formula is the key idea of our question. Let us proceed on with the solution. Now let us talk about the first part that should have exactly three girls. Now we have to have seven members in a committee to be formed from nine boys and four girls. If the committee should have exactly three girls, it can be done only by one way. That is three girls, four boys, right? So there will be as many ways of selecting three girls and there are combination of four girls taken three at a time. So the required number of ways of selecting three girls out of four, that will be four c three. Let this be the first equation. Similarly, there will be as many ways of selecting four boys, there are combination of nine boys taking four at a time. So required number of ways of selecting four boys out of nine, that will be nine c four. Let this be the second equation. So on applying the multiplication principle, the number of combinations of three girls, four boys is equal to one multiplied by the second. That is four c three multiplied by nine c four. On applying the formula, we have four factorial divided by three factorial four minus three factorial. Here we have nine factorial divided by four factorial nine minus four factorial. That is four factorial divided by three factorial multiplied by one factorial multiplied by nine factorial divided by four factorial multiplied by five factorial. 4 factorial will get cancelled out and we have 9 multiplied by 8 by 7, 6, 5 and we have 3 multiplied by 2 multiplied by 1 multiplied by 5 factorial. On simplifying we are left with 504 that is the answer to the first part. On proceeding on to the second part where the committee should have at least 3 girls. So we have 3 different cases for that or there are 2 basically ways of doing so that is the first is 3 girls and 4 boys and it should have at least 3 girls and we have 4 girls so at the match we can have 4 girls and 3 boys. So here in the first part of the question we can get that the number of combination of 3 girls and 4 boys is 4C3 multiplied by 9C4 that is equal to 504 which we found out in the first part whereas 4 girls and 3 boys the number of ways of selecting 4 girls can be 4C4 and there will be as many ways of selecting 3 boys as there are combination of 9 boys taken at the time it will be 9C3. On multiplying these 2 we have the solution as 84. So the total number of ways in which committee is formed in the second case where it should have at least 3 girls it is 504 plus 84 that is 588 and that is required answer of our second part. Let us proceed on further to the third part that is committee should have at most 3 girls that means the number of girls cannot be moved to 4 so there are 4 ways of doing that we can have 3 girls and 4 boys or we can have 2 girls and 5 boys or we can have 1 girl 6 boys we can have 0 girls or 0 girl or 7 boys. Now for the first part we can take it as directly 504 from the first part for the second part here it will be out of 4 we are taking the combination of 2 girls and out of 9 we are taking 5 boys here 4C1 and we have 9C6 and for this we have 4C we have no girls so there is no point of taking a combination we have 9C7. What we need to do over here we need to add the result of all these this we have taken from the first part and let us simplify it we have 504 plus 4C2 that will be 6 multiplied by 126 plus 4C1 multiplied by 9C6 will can be written as 4 multiplied by 84 after simplification plus 36 further 504 plus 756 plus 336 plus 36 can be written as 1632 this is our required answer we found out this value as 6 and value of this as 126 and after simplification we have written the values directly I hope you enjoyed take care bye for now