Let A={1,2,3,4,5}, B={1,2,3,4} and `f:A rarr B` is a function, then

Options

A. A. number of onto functions, if n(f(A))=4 is 240
B. B. number of onto functions, if n(f(A))=3 is 600
C. C. number of onto functions, if n(f(A))=2 is 180
D. D. number of onto functions, if n(f(A))=1 is 4

Answer

To solve the problem, we need to find the number of onto functions (surjective functions) from set A to set B, where:

- Set A = {1, 2, 3, 4, 5} (n = 5 elements)
- Set B = {1, 2, 3, 4} (m = 4 elements)

### Step 1: Understand the formula for onto functions
The number of onto functions from a set with m elements to a set with n elements is given by the formula:

\[ \text{Number of onto functions} = \sum_{r=1}^{m} (-1)^{r+1} \binom{m}{r} (m - r)^n \]

Where:
- \( m \) is the number of elements in set B (4 in this case)
- \( n \) is the number of elements in set A (5 in this case)

### Step 2: Substitute the values into the formula
Here, \( m = 4 \) and \( n = 5 \). We will compute the sum:

\[ \text{Number of onto functions} = \sum_{r=1}^{4} (-1)^{r+1} \binom{4}{r} (4 - r)^5 \]

### Step 3: Calculate each term in the sum
We will calculate the terms for \( r = 1, 2, 3, 4 \):

1. **For \( r = 1 \)**:
\[ (-1)^{1+1} \binom{4}{1} (4 - 1)^5 = 1 \cdot 4 \cdot 3^5 = 4 \cdot 243 = 972 \]

2. **For \( r = 2 \)**:
\[ (-1)^{2+1} \binom{4}{2} (4 - 2)^5 = -1 \cdot 6 \cdot 2^5 = -6 \cdot 32 = -192 \]

3. **For \( r = 3 \)**:
\[ (-1)^{3+1} \binom{4}{3} (4 - 3)^5 = 1 \cdot 4 \cdot 1^5 = 4 \cdot 1 = 4 \]

4. **For \( r = 4 \)**:
\[ (-1)^{4+1} \binom{4}{4} (4 - 4)^5 = -1 \cdot 1 \cdot 0^5 = 0 \]

### Step 4: Sum all the terms
Now we sum all the calculated values:

\[ 972 - 192 + 4 + 0 = 784 \]

### Step 5: Final calculation
The total number of onto functions from A to B is:

\[ \text{Number of onto functions} = 784 \]

### Conclusion
Thus, the number of onto functions from set A to set B is **784**.

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