Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6). Then the number of functions f: A→B satisfying f(1) + f(2) = f(4)-1 is equal to _________ .
Correct Answer: 360
Solution and Explanation
We are tasked with finding the total number of functions f that satisfy the given conditions.
Step 1: Analyze Cases for f(1) and f(2)
The sum f(1) + f(2) must satisfy f(1) + f(2) &leq 5, and both f(1) and f(2) are integers. Let's explore the possible values for f(1) and f(2):
- Case (i): f(1) = 1
If f(1) = 1, then f(2) can take values from 1, 2, 3, 4 (4 possible mappings).
- Case (ii): f(1) = 2
If f(1) = 2, then f(2) can take values from 1, 2, 3 (3 possible mappings).
- Case (iii): f(1) = 3
If f(1) = 3, then f(2) can take values from 1, 2 (2 possible mappings).
- Case (iv): f(1) = 4
If f(1) = 4, then f(2) can only take the value 1 (1 possible mapping).
Step 2: Count Mappings for f(5) and f(6)
Both f(5) and f(6) can each take any of 6 possible values independently.
Step 3: Calculate the Total Number of Functions
To compute the total number of functions, we calculate the number of ways to choose f(1), f(2), f(5), and f(6):
- For f(1) and f(2), we sum the possibilities from the cases:
(4 + 3 + 2 + 1) = 10 - For f(5) and f(6), each has 6 possible mappings:
6 × 6 = 36
Thus, the total number of functions is:
10 × 36 = 360
Final Answer:
The total number of functions is 360.
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Functions Questions
The domain of \(y= cos^{-1}|\frac{2-|x|}{4}| log(3 - x)^{-1}\) is [α, β) - {y} then the value of α+β-y =?
- If the domain of the function \(f(x)\) = \(\frac{\sqrt{x^2-25}}{(4-x^2)}+ \log(x^2+2x-15)\) is \((-∞, α) ∪ (β, ∞)\), then \(α^2+β^2\) is equal to \(b\)
- If the domain of the function \[ f(x) = \cos^{-1} \left( \frac{2 - |x|}{4} \right) + \left( \log_e (3 - x) \right)^{-1} \] is \([-\alpha, \beta) - \{ \gamma \}\), then \( \alpha + \beta + \gamma \) is equal to:
- If the function \( f(x) = \begin{cases} \frac{1}{|x|}, & |x| \geq 2 \\ ax^2 + 2b, & |x| < 2 \end{cases} \) is differentiable on \( \mathbb{R} \), then \( 48 (a + b) \) is equal to \(\_\_\_\_\_\_\_\_\_\).
- Consider the function \( f : (0, 2) \to \mathbb{R} \) defined by \[ f(x) = \frac{x}{2} + \frac{2}{x} \] and the function \( g(x) \) defined by \[ g(x) = \begin{cases} \min\{f(t)\}, & 0 < t \leq x \text{ and } 0 < x \leq 1 \\ \frac{3}{2} + x, & 1 < x < 2 \end{cases} \] Then:
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.