If `A={a,b,c}` and `B={-3,-1,0,1,3}`, then the number of injections that can be defined from A to B is .

Options

A. 125
B. 243
C. 60
D. 120

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 1: Identify the sets
We have two sets:
- Set A = {a, b, c} (which has 3 elements)
- Set B = {-3, -1, 0, 1, 3} (which has 5 elements)

### Step 2: Understand the concept of injections
An injection (or one-to-one function) from set A to set B 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.

### Step 3: Determine the number of choices for each element in A
1. For the first element in A (let's say 'a'), we can choose any of the 5 elements from B. So, there are 5 choices.
2. For the second element in A (let's say 'b'), we can choose from the remaining elements in B. Since one element has already been used for 'a', we have 4 choices left.
3. For the third element in A (let's say 'c'), we can choose from the remaining elements in B. Since two elements have already been used for 'a' and 'b', we have 3 choices left.

### Step 4: Calculate the total number of injections
To find the total number of injections, we multiply the number of choices for each element:
\[ \text{Total injections} = 5 \times 4 \times 3 \]

### Step 5: Perform the calculation
Now, we calculate:
\[ 5 \times 4 = 20 \]
\[ 20 \times 3 = 60 \]

### Conclusion
The total number of injections that can be defined from set A to set B is **60**.