`(1 - (1)/(3)) (1 - (1)/(4)) (1 - (1)/(5)) ….. (1 - (1)/(25))` is equal to

Options

A. `(2)/(25)`
B. `(1)/(25)`
C. `1"" (19)/(25)`
D. `(1)/(325)`

Answer

To solve the expression \( (1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{25}) \), we will break it down step by step.

### Step 1: Rewrite each term
We can rewrite each term in the product:
\[ 1 - \frac{1}{n} = \frac{n-1}{n} \]
Thus, the expression becomes:
\[ (1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{25}) = \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{24}{25} \]

### Step 2: Write the product explicitly
Now, we can write the product explicitly:
\[ \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{24}{25} \]

### Step 3: Observe cancellation
Notice that in this product, all the intermediate terms will cancel out. Specifically, the numerator of each fraction cancels with the denominator of the next fraction:
\[ = \frac{2 \cdot 3 \cdot 4 \cdots \cdot 24}{3 \cdot 4 \cdots \cdot 25} \]

### Step 4: Simplify the expression
After cancellation, we are left with:
\[ = \frac{2}{25} \]

### Final Answer
Thus, the value of the expression \( (1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{25}) \) is:
\[ \frac{2}{25} \]