a) Four and five digit numbers greater than 6000 are obtained by arranging the digits 3, 4, 5, 6, 7. How many of these are i) Odd numbers (Ans: 30) ii) Even numbers (Ans 48) b) Nine out of twelve members of the school’s Geography club are to be taken out on a study tour. Given that there are seven boys and five girls and at least three girls have to go for the tour. (i) Find the number of ways in which the selection of students can be done (Ans: 16 ways) (ii) If there are two sisters in the club, who are definitely selected to go, in how many ways can the remaining students be selected? (Ans: 161 ways)

Step 1: Understand the problem for part a). We are arranging the digits 3, 4, 5, 6, 7 to form four-digit and five-digit numbers greater than 6000. The question provides answers for i) and ii) that correspond to specific cases (four-digit for i) and five-digit for ii)). We will proceed by calculating for these specific cases to match the provided answers. a) i) Odd numbers We assume this refers to four-digit odd numbers greater than 6000, to match the given answer of 30. The digits available are \3, 4, 5, 6, 7\. A four-digit number $d_1