Find n If :
` ""^(n)P_4 : "" ^(n-1)P_3 = 9: 1`

Answer

To solve the problem, we need to find the value of \( n \) given the ratio of permutations:

\[ \frac{nP_4}{(n-1)P_3} = \frac{9}{1} \]

### Step 1: Write the permutation formulas

The formula for permutations is given by:

\[ nP_r = \frac{n!}{(n-r)!} \]

So, we can express \( nP_4 \) and \( (n-1)P_3 \) using this formula:

\[ nP_4 = \frac{n!}{(n-4)!} \]
\[ (n-1)P_3 = \frac{(n-1)!}{((n-1)-3)!} = \frac{(n-1)!}{(n-4)!} \]

### Step 2: Substitute the formulas into the ratio

Now substitute these expressions into the ratio:

\[ \frac{nP_4}{(n-1)P_3} = \frac{\frac{n!}{(n-4)!}}{\frac{(n-1)!}{(n-4)!}} \]

### Step 3: Simplify the ratio

The \( (n-4)! \) cancels out:

\[ \frac{n!}{(n-1)!} \]

Using the property of factorials, we know that:

\[ n! = n \cdot (n-1)! \]

So we can rewrite the ratio as:

\[ \frac{n \cdot (n-1)!}{(n-1)!} = n \]

### Step 4: Set up the equation

Now we have:

\[ n = \frac{9}{1} = 9 \]

### Step 5: Conclusion

Thus, the value of \( n \) is:

\[ \boxed{9} \]