In a class of 400students, 150are interested in doing a statistics pro

1.	In a class of 400students, 150are interested in doing a statistics project, 300are interested in doing a machine learning project and 25are not interested in doing either of the projects. Then the number of students who are interested in doing both the projects, is
Answer» Given: Total number of students, n(U) = 400 Number of students in Statistics Project, n(A) = 150 Number of students in Machine Learning Project, n(B) = 300 Number of students not in any Project, n(A $\cup$ B)' = 25 To FIND: Number of students in both the projects, n(A $\cap$ B) = ? SOLUTION: We know the formula: $n(A\cup B) = n(A) +n(B)-n(A\cap B)$ To find $n(A\cap B)$ we need $n(A \cup B)$ . We know that total number of students is equal to the sum of students in at LEAST ONE project and student not in any project. i.e. $n(U) = n(A\cup B)+n(A\cup B)'\\\Rightarrow 400 = n(A\cup B)+25\\\Rightarrow n(A\cup B)=375$ Now, putting the values in the above formula: $375 = 150+300-n(A\cap B)\\\Rightarrow 375 = 450-n(A\cap B)\\\Rightarrow n(A\cap B)= 450-375 \\\Rightarrow n(A\cap B)= 75$ So, answer is: Number of students in both the projects, n(A $\cap$ B) = 75