Out of 80 students who appeared for the school exams in Mathematics (M), Physics (P) and Chemistry (C), 50 passed M, 30 passed P and 40 passed C. At most 20 students passed M and P, at most 20 students passed P and C and at most 20 students passed C and M. The maximum number of students who could have passed all three exams is ____________.

In a four-digit number, the product of thousands digit and units digit is zero while their difference is 7. Product of the middle digits is 18. The thousands digit is as much more than the units digit as the hundreds digit is more than the tens digit. The four-digit number is ___…

Two friends run a 3 kilometer race along a circular course of length 300 meters. If their speeds are in ratio 3:2, the number of times the winner passes the other is ____________.

Ashok purchased pens and pencils in the ratio 2:3 during his first visit and paid Rs. 86 to the shopkeeper. During his second visit, he purchased pens and pencils in the ratio 4:1 and paid Rs. 112. The cost of a pen as well as a pencil in rupees is a positive integer. If Ashok pu…

Out of 13 objects, 4 are indistinguishable and rest are distinct. The number of ways we can choose 4 objects out of 13 objects is ____________.

The minimum value of \(f(x) = |3 - x| + |2 + x| + |5 - x|\) is equal to ____________.

The probability that a randomly chosen factor of \(10^{19}\) is a multiple of \(10^{15}\) is

Suppose \(\begin{vmatrix} a & a^2 & a^3-1 \\ b & b^2 & b^3-1 \\ c & c^2 & c^3-1 \end{vmatrix} = 0\), where a, b and c are distinct real numbers. If a = 3, then the value of abc is ____________.

The number of acute angled triangles whose sides are three consecutive positive integers and whose perimeter is at most 100 is