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 can use the formula for the number of onto functions, which is given by:

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

where:
- \( m \) is the number of elements in set \( A \),
- \( n \) is the number of elements in set \( B \),
- \( r \) is the variable that runs from \( 1 \) to \( n \).

### Step 1: Identify the values of \( m \) and \( n \)

For our sets:
- Set \( A \) has 4 elements, so \( m = 4 \).
- Set \( B \) has 3 elements, so \( n = 3 \).

### Step 2: Substitute \( m \) and \( n \) into the formula

Now we substitute \( m \) and \( n \) into the formula:

\[ \text{Number of onto functions} = \sum_{r=1}^{3} (-1)^{3-r} \binom{3}{r} r^4 \]

### Step 3: Calculate the summation for each value of \( r \)

We need to evaluate the summation for \( r = 1, 2, 3 \).

#### For \( r = 1 \):

\[ (-1)^{3-1} \binom{3}{1} 1^4 = (-1)^2 \cdot 3 \cdot 1 = 3 \]

#### For \( r = 2 \):

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

#### For \( r = 3 \):

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

### Step 4: Combine the results

Now we combine the results from all three calculations:

\[ \text{Total} = 3 - 48 + 81 = 36 \]

### Conclusion

Thus, the total number of onto functions from the set \( A \) to the set \( B \) is \( 36 \).

### Final Answer

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

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