Set A has 3 elements and the set B has 4 elements. Then, the number of injective mappings that can be defined from A to B is :

Options

A. 144
B. 12
C. 24
D. 64

Answer

To find the number of injective mappings (one-to-one functions) from set A to set B, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Elements of Sets A and B**:
- Let set A have 3 elements: \( A = \{ a_1, a_2, a_3 \} \).
- Let set B have 4 elements: \( B = \{ b_1, b_2, b_3, b_4 \} \).

2. **Understanding Injective Mappings**:
- An injective mapping means that each element in set A must map to a unique element in set B. No two elements in A can map to the same element in B.

3. **Choosing Mappings for Each Element**:
- For the first element \( a_1 \) in set A, we can choose any of the 4 elements in set B. So, there are 4 choices for \( a_1 \).
- After choosing an element for \( a_1 \), we have 3 elements left in set B for the second element \( a_2 \). Thus, there are 3 choices for \( a_2 \).
- After choosing elements for \( a_1 \) and \( a_2 \), we have 2 elements left in set B for the third element \( a_3 \). Therefore, there are 2 choices for \( a_3 \).

4. **Calculating the Total Number of Injective Mappings**:
- The total number of injective mappings can be calculated by multiplying the number of choices for each element:
\[ \text{Total Injective Mappings} = 4 \times 3 \times 2 \]

5. **Performing the Calculation**:
\[ 4 \times 3 = 12 \]
\[ 12 \times 2 = 24 \]

6. **Conclusion**:
- Therefore, the total number of injective mappings from set A to set B is \( 24 \).

### Final Answer:
The number of injective mappings that can be defined from A to B is **24**.

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