Evalute `(n!)/((n-r)!)`. when `n=20` and `r=2`.

Answer

To evaluate the expression \(\frac{n!}{(n-r)!}\) when \(n=20\) and \(r=2\), we can follow these steps:

### Step 1: Identify the values of \(n\) and \(r\)
We are given:
- \(n = 20\)
- \(r = 2\)

### Step 2: Substitute the values into the expression
We need to evaluate:
\[ \frac{n!}{(n-r)!} = \frac{20!}{(20-2)!} = \frac{20!}{18!} \]

### Step 3: Simplify the factorial expression
Using the property of factorials, we can express \(20!\) as:
\[ 20! = 20 \times 19 \times 18! \]
Thus, we can rewrite our expression:
\[ \frac{20!}{18!} = \frac{20 \times 19 \times 18!}{18!} \]

### Step 4: Cancel out the common terms
The \(18!\) in the numerator and denominator cancels out:
\[ \frac{20 \times 19 \times 18!}{18!} = 20 \times 19 \]

### Step 5: Calculate the final result
Now, we compute:
\[ 20 \times 19 = 380 \]

### Final Answer:
The value of \(\frac{n!}{(n-r)!}\) when \(n=20\) and \(r=2\) is \(380\).

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