Let \( R = \{a, b, c, d, e\} \) and \( S = \{1, 2, 3, 4\} \). Total number of onto functions \( f: R \to S \) such that \( f(a) \neq 1 \), is equal to:
Let \( R = \{a, b, c, d, e\} \) and \( S = \{1, 2, 3, 4\} \). Total number of onto functions \( f: R \to S \) such that \( f(a) \neq 1 \), is equal to:
Show Hint
When counting onto functions with restrictions, calculate the total onto functions first and subtract the restricted cases using inclusion-exclusion principles.
- 240
- 180
- 204
- 216
The Correct Option is B
Approach Solution - 1
The correct answer is (B) : 180
Total no. of onto functions
\(=\frac{5!}{3!2!}\times4!\)
So , when f(a) = 1
\(\frac{4!}{2!2!}\times3!+4!\)
\(\therefore\) Required functions :
= 240 -36 -24
=180
Approach Solution -2
The total number of onto functions from \( R \) to \( S \) is calculated as:
\[ \text{Total onto functions} = \binom{5}{3} \cdot 4! = \frac{5 \cdot 4}{2} \cdot 24 = 240. \]
Now, consider the case where \( f(a) = 1 \).
If \( f(a) = 1 \), the remaining 4 elements \( b, c, d, e \) must map onto \( S \setminus \{1\} \), which has 3 elements. The number of onto functions for these remaining 4 elements is:
\[ \text{Functions with \( f(a) = 1 \)} = \binom{4}{2} \cdot 3! \cdot 3. \]
Compute this step by step:
\[ \binom{4}{2} \cdot 3! = \frac{4 \cdot 3}{2} \cdot 6 +14= 60. \]
Finally, subtract this from the total:
\[ \text{Required functions} = 240 - 60 = 180. \]
Thus, the total number of onto functions \( f \) such that \( f(a) \neq 1 \) is \( \boxed{180} \).
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 Relations and functions Questions
- If the domain of the function \( f(x) = \log_e \left( \frac{2x + 3}{4x^2 + x - 3} \right) + \cos^{-1} \left( \frac{2x - 1}{x + 2} \right) \) is \( (\alpha, \beta] \), then the value of \( 5\beta - 4\alpha \) is equal to
- If \(f(x) = (x - 2)^2 (x - 3)^3\) and \(x ∈ [1, 4]\) and If \(M\) and \(m\) denotes maximum and minimum values respectively, then \(M - m\) is
- If \(f(x) =ln(\frac {1-x^2}{1+x^2})\) then value of \(225(f'(x) – f''(x))\) at \(x=\frac 12\)
- Let \( A = \{ 1, 2, 3, \dots, 20 \} \). Let \( R_1 \) and \( R_2 \) be two relations on \( A \) such that \(R_1 = \{(a, b) : b \text{ is divisible by } a\}\)
and \(R_2 = \{(a, b) : a \text{ is an integral multiple of } b\}\).Then, the number of elements in \( R_1 - R_2 \) is equal to \(\_\_\_\_.\) - Let \( A = \{1, 2, 3, 4\} \) and \( R = \{(1, 2), (2, 3), (1, 4)\} \) be a relation on \( A \).Let \( S \) be the equivalence relation on \( A \) such that \( R \subseteq S \) and the number of elements in \( S \) is \( n \). Then, the minimum value of \( n \) is _____
Top JEE Main Questions
- 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 ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
