Statement-1: If two sets X and Y contain 3 and 5 elements respectively, then `.^(5)C_(3)xx3!` one-one functions can be defined from X to Y.
Statement:2: A one-one function from X to Y relates different element of set X to different elements of set Y.

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to evaluate the two statements provided regarding one-one functions between two sets, X and Y.

### Step-by-Step Solution:

**Step 1: Analyze Statement 1**
- We have two sets: Set X with 3 elements and Set Y with 5 elements.
- We need to find the number of one-one functions from X to Y.

**Step 2: Understanding One-One Functions**
- A one-one function (or injective function) means that each element in set X must map to a unique element in set Y.
- Since set X has 3 elements, we need to choose 3 unique elements from set Y (which has 5 elements) to map to.

**Step 3: Calculate the Number of Ways to Choose Elements**
- The number of ways to choose 3 elements from 5 is given by the combination formula \( \binom{n}{r} \), which is \( \binom{5}{3} \).
- This can be calculated as:
\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

**Step 4: Calculate the Number of Arrangements**
- Once we have chosen 3 elements from set Y, we can arrange these 3 elements in \( 3! \) (factorial of 3) ways.
- This can be calculated as:
\[ 3! = 3 \times 2 \times 1 = 6 \]

**Step 5: Total Number of One-One Functions**
- The total number of one-one functions from X to Y is the product of the number of ways to choose the elements and the number of arrangements:
\[ \text{Total one-one functions} = \binom{5}{3} \times 3! = 10 \times 6 = 60 \]

**Conclusion for Statement 1:**
- Since we have calculated the total number of one-one functions from X to Y as 60, Statement 1 is **True**.

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**Step 6: Analyze Statement 2**
- Statement 2 claims that a one-one function from X to Y relates different elements of set X to different elements of set Y.
- By definition, a one-one function means that each element in set X is related to exactly one unique element in set Y.

**Conclusion for Statement 2:**
- The statement is misleading because it suggests that elements of X can relate to multiple elements in Y, which contradicts the definition of a one-one function. Therefore, Statement 2 is **False**.

### Final Answer:
- Statement 1: True
- Statement 2: False

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