The total number of onto functions from the set {1,2,3,4) to the set (3,4,7) is

Options

A. 18
B. 36
C. 64
D. none of these

Explanation

To find the total number of onto functions from the set \( A = \{1, 2, 3, 4\} \) to the set \( B = \{3, 4, 7\} \), we will follow these steps:

### Step 1: Identify the cardinalities of the sets
- The cardinality of set \( A \) (denoted as \( m \)) is 4 (since it has 4 elements).
- The cardinality of set \( B \) (denoted as \( n \)) is 3 (since it has 3 elements).

### Step 2: Check if onto functions can exist
- An onto function (or surjective function) from set \( A \) to set \( B \) exists if \( m \geq n \). Here, since \( 4 \geq 3 \), onto functions can exist.

### Step 3: Use the formula for the number of onto functions
The number of onto functions from a set of \( m \) elements to a set of \( n \) elements is given by the formula:
\[ \text{Number of onto functions} = \sum_{r=1}^{n} (-1)^{n-r} \binom{n}{r} r^m \]
Where \( \binom{n}{r} \) is the binomial coefficient.

### Step 4: Substitute the values into the formula
Here, \( m = 4 \) and \( n = 3 \):
\[ \text{Number of onto functions} = \sum_{r=1}^{3} (-1)^{3-r} \binom{3}{r} r^4 \]

### Step 5: Calculate each term in the summation
1. For \( r = 1 \):
\[ (-1)^{3-1} \binom{3}{1} 1^4 = (-1)^2 \cdot 3 \cdot 1 = 3 \]

2. For \( r = 2 \):
\[ (-1)^{3-2} \binom{3}{2} 2^4 = (-1)^1 \cdot 3 \cdot 16 = -48 \]

3. For \( r = 3 \):
\[ (-1)^{3-3} \binom{3}{3} 3^4 = (-1)^0 \cdot 1 \cdot 81 = 81 \]

### Step 6: Combine the results
Now, we sum the results from each term:
\[ 3 - 48 + 81 = 36 \]

### Final Answer
The total number of onto functions from the set \( \{1, 2, 3, 4\} \) to the set \( \{3, 4, 7\} \) is **36**.

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