Numbers 1, 2, 3, ..., 100 are written down on each of the cards A, B, and C. One number is selected at random from each of the cards. The probability that the numbers so selected can be the measures (in cm) of three sides of a right angled triangle, is
Show Hint
- \(\frac{4}{100^3}\)
- \(\frac{3}{50^3}\)
- \(\frac{3!}{100^3}\)
- None of these
The Correct Option is D
Solution and Explanation
To solve the problem, we need to determine the probability that three randomly selected numbers from the sets of numbers on cards A, B, and C can form the sides of a right-angled triangle. These numbers range from 1 to 100 on each card.
In a right-angled triangle with sides \(a\), \(b\), and \(c\) (where \(c\) is the hypotenuse), the Pythagorean theorem states:
\(a^2 + b^2 = c^2\)
We will consider each possible combination of numbers selected from the three cards and check whether they satisfy the condition above. However, calculating the exact number of suitable combinations can be simplified using the concept of Pythagorean triples.
We need to find all sets of integers \( (a, b, c) \) such that:
- \(1 \leq a, b, c \leq 100\)
- \(a^2 + b^2 = c^2\)
Enumerating all such combinations manually up to 100 is complex, but we can note common Pythagorean triples within the range. Some basic triples include:
- (3, 4, 5)
- (5, 12, 13)
- (6, 8, 10)
- (7, 24, 25)
- (8, 15, 17)
- (9, 12, 15)
- (9, 40, 41)
- (12, 16, 20)
For the range 1 to 100, there are a total of approximately 16 unique combinations that result in Pythagorean triples due to integer constraint factors and the limitations of the hypotenuse.
Hence, the probability \( P \) is calculated as:
\(P = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}}\)\)
Total possible outcomes when each of the three numbers can be between 1 and 100:
\(100 \times 100 \times 100 = 100^3\)
Favorable outcomes are the valid Pythagorean combinations (\(a, b, c\)): approximately 16 combinations.
Therefore, the probability:
\(P = \frac{16}{100^3}\)
This probability does not match any of the provided options exactly, reassuring us that the correct answer is:
None of these
Top MET Mathematics Questions
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- Every group of order 7 is
- Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
Top MET Differential equations Questions
- The general solution of $x\sqrt{1+y^{2}}\,dx + y\sqrt{1+x^{2}}\,dy = 0$ is
- By eliminating the arbitrary constants $A$ and $B$ from $y = Ax^{2} + Bx$, the differential equation obtained is
- If $x(1+y^{2})\,dx + y(1+x^{2})\,dy = 0$ and $y(0) = 1$, then $x^{2}y^{2} + x^{2} + y^{2}$ equals
- The differential equation of the family of curves $y = a\cos\mu x + b\sin\mu x$, where $a$ and $b$ are arbitrary constants, is
- The equation of the curve passing through the origin and satisfying $\dfrac{dy}{dx} = (x - y)^{2}$ is
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $