Evaluate `(n!)/((n-r)!)` when n = 12 and r = 3.

Answer

To evaluate the expression \(\frac{n!}{(n-r)!}\) when \(n = 12\) and \(r = 3\), follow these steps:

### Step 1: Substitute the values of \(n\) and \(r\)
We start by substituting \(n = 12\) and \(r = 3\) into the expression:
\[ \frac{n!}{(n-r)!} = \frac{12!}{(12-3)!} = \frac{12!}{9!} \]

### Step 2: Expand \(12!\)
Next, we can express \(12!\) in terms of \(9!\):
\[ 12! = 12 \times 11 \times 10 \times 9! \]

### Step 3: Substitute back into the expression
Now we substitute this expansion back into our expression:
\[ \frac{12!}{9!} = \frac{12 \times 11 \times 10 \times 9!}{9!} \]

### Step 4: Simplify the expression
The \(9!\) in the numerator and denominator cancels out:
\[ = 12 \times 11 \times 10 \]

### Step 5: Calculate the product
Now, we calculate the product:
\[ 12 \times 11 = 132 \]
\[ 132 \times 10 = 1320 \]

### Final Answer
Thus, the value of \(\frac{n!}{(n-r)!}\) when \(n = 12\) and \(r = 3\) is:
\[ \boxed{1320} \]