Let `n(A)=3 and n(B)=5`, then the number of one - one functions from A to B is

Options

A. 15
B. 60
C. 125
D. 10

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 number of elements in each set
Given:
- \( n(A) = 3 \) (the number of elements in set A)
- \( n(B) = 5 \) (the number of elements in set B)

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

### Step 3: Calculate the number of choices for each element in A
1. The first element in A can be mapped to any of the 5 elements in B.
2. The second element in A can then be mapped to any of the remaining 4 elements in B (since one element is already taken).
3. The third element in A can be mapped to any of the remaining 3 elements in B.

### Step 4: Calculate the total number of one-to-one functions
The total number of one-to-one functions from A to B can be calculated by multiplying the number of choices for each element in A:
\[ \text{Total one-to-one functions} = n(B) \times (n(B) - 1) \times (n(B) - 2) \]
Substituting the values:
\[ = 5 \times 4 \times 3 \]

### Step 5: Perform the multiplication
Calculating:
\[ 5 \times 4 = 20 \]
\[ 20 \times 3 = 60 \]

### Final Answer
The number of one-to-one functions from set A to set B is **60**.

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