In this unit, we have learned about the fundamental concepts of combinatorics, which is the branch of mathematics that deals with counting and arranging objects. Combinatorics is a crucial field of study with many real-world applications. For this discussion, two scenarios are given. Answering both parts is compulsory. Assume you are the principal of the host school. Part 1: You are organizing a swimming competition with 'n' participants. The school has announced scholarships for the top three swimmers. The details of the scholarship is as follows: First place - 100% scholarship to study abroad Second-place - 100 % scholarship to study in any school all over the country Third-place - 50% scholarship to study in any school all over the country Take 'n' as any two-digit number representing the total number of participants in the competition. Find how many possible ways are there to award these scholarships if there are no ties during the competition. Explain in detail. Part 2: A group of n-school principals are trained to implement innovative education in their schools. (Choose a two-digit number for n). As a part of the innovative education scheme, 9 schools are to be offered equal funds from the government. i) In how many ways can 9 schools be selected from the trained group? ii) If your school is one among the 9 selected schools, how many ways will be left to choose the other schools? In both parts 1 and 2, which concepts are you using? Explain the reason for choosing the concept. Also, discuss how combinatorics helps to solve real-world counting problems. Reference: Levin, O. (2021). Discrete mathematics: An open introduction (3rd ed.). https://discrete.openmathbooks.org/dmoi3/frontmatter.html licensed under CC 4.0