If `A= {1,3,5,7} and B= {1,2,3,4,5,6,7,8}`, then the number of one-to-one functions from A into B is

Options

A. 1340
B. 1680
C. 1430
D. 1880

Answer

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

### Step 1: Identify the elements in sets A and B
- Set A = {1, 3, 5, 7} has 4 elements.
- Set B = {1, 2, 3, 4, 5, 6, 7, 8} has 8 elements.

### Step 2: Understand the concept of one-to-one functions
A one-to-one function (injective function) from set A to set B means that each element in A maps to a unique element in B. Therefore, no two elements in A can map to the same element in B.

### Step 3: Use the formula for the number of one-to-one functions
The number of one-to-one functions from a set with \( m \) elements to a set with \( n \) elements is given by the formula:
\[ nP_m = \frac{n!}{(n-m)!} \]
where \( nP_m \) is the number of permutations of \( n \) items taken \( m \) at a time.

### Step 4: Apply the formula
In our case:
- \( m = 4 \) (the number of elements in set A)
- \( n = 8 \) (the number of elements in set B)

So we need to calculate \( 8P4 \):
\[ 8P4 = \frac{8!}{(8-4)!} = \frac{8!}{4!} \]

### Step 5: Calculate \( 8P4 \)
Calculating \( 8! \) and \( 4! \):
- \( 8! = 8 \times 7 \times 6 \times 5 \times 4! \)
- Therefore, \( 8P4 = 8 \times 7 \times 6 \times 5 \)

### Step 6: Perform the multiplication
Now we calculate:
\[ 8 \times 7 = 56 \]
\[ 56 \times 6 = 336 \]
\[ 336 \times 5 = 1680 \]

### Final Answer
The number of one-to-one functions from set A to set B is **1680**.

---