Set A has 3 elements and set B has 4 elements. The number of injections that can be defined from A to B is

Options

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

Answer

To find the number of injections (one-to-one functions) that can be defined from set A to set B, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Elements in 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. **Understand the Concept of Injection**:
- An injection (or one-to-one function) 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. **Choose an Element from Set B for Each Element in Set A**:
- For the first element \( a_1 \) in set A, we can choose any of the 4 elements from set B. So, there are 4 choices for \( a_1 \).
- After choosing an element for \( a_1 \), we cannot use that element again for \( a_2 \). Therefore, for \( a_2 \), we have 3 remaining choices.
- Finally, for \( a_3 \), we will have 2 choices left, as two elements from set B have already been used.

4. **Calculate the Total Number of Injections**:
- The total number of injections can be calculated by multiplying the number of choices for each element:
\[ \text{Total Injections} = 4 \times 3 \times 2 \]

5. **Perform the Calculation**:
- Calculate \( 4 \times 3 = 12 \)
- Then, \( 12 \times 2 = 24 \)

6. **Conclusion**:
- Therefore, the number of injections that can be defined from set A to set B is \( 24 \).

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