When simplified the product
`(1 - 1/2) (1 - 1/3) (1 - 1/4) .........(1-1/n)` gives :

Options

A. `1/n`
B. `2/n`
C. `(2(n-1))/(n)`
D. `(2)/(n (n - 1))`

Answer

To simplify the product \( (1 - \frac{1}{2})(1 - \frac{1}{3})(1 - \frac{1}{4}) \ldots (1 - \frac{1}{n}) \), we can follow these steps:

### Step 1: Rewrite each term
Each term in the product can be rewritten as:
\[ 1 - \frac{1}{k} = \frac{k - 1}{k} \]
So we can express the entire product as:
\[ (1 - \frac{1}{2})(1 - \frac{1}{3})(1 - \frac{1}{4}) \ldots (1 - \frac{1}{n}) = \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdots \frac{n-1}{n} \]

### Step 2: Write the product in fraction form
Now, we can write the product as a single fraction:
\[ \frac{1 \cdot 2 \cdot 3 \cdots (n-1)}{2 \cdot 3 \cdots n} \]

### Step 3: Simplify the fraction
Notice that in the numerator, we have \( (n-1)! \) (the factorial of \( n-1 \)), and in the denominator, we can express it as:
\[ 2 \cdot 3 \cdots n = n! / 1 \]
Thus, we can rewrite the fraction as:
\[ \frac{(n-1)!}{n!} \]

### Step 4: Further simplify
Using the property of factorials, we know that:
\[ n! = n \cdot (n-1)! \]
So we can substitute this into our expression:
\[ \frac{(n-1)!}{n \cdot (n-1)!} = \frac{1}{n} \]

### Conclusion
Therefore, the simplified product is:
\[ \frac{1}{n} \]

### Final Answer
The final answer is \( \frac{1}{n} \).