If `.^(n+2)C_(8) : ^(n-2)P_(4)=57:16`, then the value of `(n)/(2)` is

Answer

To solve the problem, we need to find the value of \( \frac{n}{2} \) given the equation:

\[ \frac{{^{(n+2)}C_8}}{{^{(n-2)}P_4}} = \frac{57}{16} \]

### Step 1: Write the formulas for combinations and permutations

The combination \( ^{n}C_r \) is given by:

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

The permutation \( ^{n}P_r \) is given by:

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

### Step 2: Substitute the values into the equation

Substituting \( n+2 \) and \( n-2 \) into the formulas, we have:

\[ ^{(n+2)}C_8 = \frac{(n+2)!}{8!(n-6)!} \]

\[ ^{(n-2)}P_4 = \frac{(n-2)!}{(n-6)!} \]

### Step 3: Set up the equation

Now substituting these into the ratio, we get:

\[ \frac{\frac{(n+2)!}{8!(n-6)!}}{\frac{(n-2)!}{(n-6)!}} = \frac{57}{16} \]

### Step 4: Simplify the equation

The \( (n-6)! \) cancels out:

\[ \frac{(n+2)!}{8!(n-2)!} = \frac{57}{16} \]

### Step 5: Rewrite \( (n+2)! \)

Using the property of factorials, we can rewrite \( (n+2)! \):

\[ (n+2)! = (n+2)(n+1)(n)(n-1)(n-2)! \]

Substituting this back into the equation gives:

\[ \frac{(n+2)(n+1)(n)(n-1)(n-2)!}{8!(n-2)!} = \frac{57}{16} \]

### Step 6: Cancel out \( (n-2)! \)

Now we can cancel \( (n-2)! \):

\[ \frac{(n+2)(n+1)(n)(n-1)}{8!} = \frac{57}{16} \]

### Step 7: Multiply both sides by \( 8! \)

Multiplying both sides by \( 8! \) gives:

\[ (n+2)(n+1)(n)(n-1) = \frac{57}{16} \cdot 8! \]

### Step 8: Calculate \( 8! \)

Calculating \( 8! \):

\[ 8! = 40320 \]

Thus:

\[ \frac{57}{16} \cdot 40320 = 57 \cdot 2520 = 143640 \]

### Step 9: Set up the equation

Now we have:

\[ (n+2)(n+1)(n)(n-1) = 143640 \]

### Step 10: Solve for \( n \)

We can use trial and error to find \( n \). Testing \( n = 19 \):

\[ (19+2)(19+1)(19)(19-1) = 21 \cdot 20 \cdot 19 \cdot 18 \]

Calculating this:

\[ 21 \cdot 20 = 420 \]
\[ 420 \cdot 19 = 7980 \]
\[ 7980 \cdot 18 = 143640 \]

Since this is correct, we find \( n = 19 \).

### Step 11: Find \( \frac{n}{2} \)

Now we calculate:

\[ \frac{n}{2} = \frac{19}{2} = 9.5 \]

### Final Answer

Thus, the value of \( \frac{n}{2} \) is:

\[ \boxed{9.5} \]