1. **State the problem:** We have a group of students where 30 play cricket, 38 play football, and 9 play neither cricket nor football. We need to find the lowest possible number of students in the group. 2. **Formula and rules:** Let $C$ be the set of students who play cricket, $F$ be the set of students who play football, and $N$ be the number who play neither. The total number of students $T$ is given by: $$ T = |C \cup F| + N $$ Using the principle of inclusion-exclusion: $$ |C \cup F| = |C| + |F| - |C \cap F| $$ To minimize $T$, we want to maximize the overlap $|C \cap F|$ because the more students who play both sports, the fewer total students there are. 3. **Calculate maximum overlap:** The maximum overlap is the smaller of the two groups: $$ |C \cap F|_{max} = \min(|C|, |F|) = \min(30, 38) = 30 $$ 4. **Calculate lowest total number of students:** $$ T_{min} = |C| + |F| - |C \cap F|_{max} + N = 30 + 38 - 30 + 9 = 47 $$ 5. **Interpretation:** The lowest possible number of students in the group is 47.