Here are the solutions to the questions: 1. Set Theory Problem Given: Total students, $n(U) = 120$ Students who liked cricket, $n(C) = 60$ Students who liked basketball, $n(B) = 55$ Students who did not like any of the two games, $n(C \cup B)' = 20$ a) If C and B denote the sets of students who liked cricket and basketball game respectively, write the cardinality of $n(B \cup C)$. We know that $n(U) = n(C \cup B) + n(C \cup B)'$. $$120 = n(C \cup B) + 20$$ $$n(C \cup B) = 120 - 20$$ $$n(C \cup B) = 100$$ The cardinality of $n(B \cup C)$ is $\boxed{100}$. b) Present the above information in a Venn-diagram. First, find the number of students who liked both games, $n(C \cap B)$. We use the formula: $n(C \cup B) = n(C) + n(B) - n(C \cap B)$. $$100 = 60 + 55 - n(C \cap B)$$ $$100 = 115 - n(C \cap B)$$ $$n(C \cap B) = 115 - 100$$ $$n(C \cap B) = 15$$ Now, find the number of students who liked cricket only, $n_0(C)$. $$n_0(C) = n(C) - n(C \cap B) = 60 - 15 = 45$$ And the number of students who liked basketball only, $n_0(B)$. $$n_0(B) = n(B) - n(C \cap B) = 55 - 15 = 40$$ The Venn diagram is as follows: (Imagine two overlapping circles. The left circle represents Cricket (C), the right circle represents Basketball (B). The overlapping region is $C \cap B$. The rectangle enclosing both circles represents the Universal set (U).) • In the region for C only: 45 • In the region for B only: 40 • In the intersection region ($C \cap B$): 15 • Outside both circles (neither liked): 20 c) Find the number of students who liked cricket only. The number of students who liked cricket only is $n_0(C)$. $$n_0(C) = n(C) - n(C \cap B) = 60 - 15 = 45$$ The number of students who liked cricket only is $\boxed{45}$. d) Compare the number of students who liked both cricket and basketball and who liked except these two games. Number of students who liked both cricket and basketball, $n(C \cap B) = 15$. Number of students who liked neither of the two games, $n(C \cup B)' = 20$. Comparing these two values: $20 > 15$. Therefore, the number of students who liked neither game is greater than the number of students who liked both games. 2. Simple and Compound Interest Problem Given: Principal, $P = \text{Rs. } 60,000$ Time, $T = 2$ years Rate, $R = 10\%$ p.a. a) Define compound interest. Compound interest is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. It is often referred to as "interest on interest." b) Find the compound interest received by Bijaya after 2 years. Step 1: Calculate the amount ($A$) using the compound interest formula.