The number of one-one functions from a set containing 2 elements to a set containing 3 elements is:

Options

A. 2
B. 3
C. 6
D. 4

Answer

To find the number of one-one functions from a set containing 2 elements to a set containing 3 elements, we can follow these steps:

### Step 1: Understand the concept of one-one functions
A one-one function (or injective function) means that every element in the domain (the set with 2 elements) maps to a unique element in the codomain (the set with 3 elements). This means no two elements in the domain can map to the same element in the codomain.

### Step 2: Identify the sets
Let’s denote the domain set as \( A = \{ a_1, a_2 \} \) (which contains 2 elements) and the codomain set as \( B = \{ b_1, b_2, b_3 \} \) (which contains 3 elements).

### Step 3: Choose elements from the codomain
Since we have 2 elements in the domain and we want to assign them to 3 elements in the codomain uniquely, we can choose 2 elements from the codomain set \( B \). The number of ways to choose 2 elements from 3 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of elements in the codomain and \( r \) is the number of elements we want to choose.

\[ \text{Number of ways to choose 2 elements from 3} = \binom{3}{2} = 3 \]

### Step 4: Arrange the chosen elements
After choosing 2 elements from the codomain, we can arrange these 2 elements in \( 2! \) (factorial of 2) ways. This is because each element in the domain can be mapped to any of the chosen elements in the codomain.

\[ \text{Number of arrangements of 2 elements} = 2! = 2 \]

### Step 5: Calculate the total number of one-one functions
To find the total number of one-one functions, we multiply the number of ways to choose the elements by the number of arrangements:

\[ \text{Total number of one-one functions} = \binom{3}{2} \times 2! = 3 \times 2 = 6 \]

### Conclusion
Thus, the number of one-one functions from a set containing 2 elements to a set containing 3 elements is **6**.

---