If ` ""^(4)P_2= n.""^(4)C_2.` Find the value on f.

Answer

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

\[ 4P2 = n \cdot 4C2 \]

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

The formula for permutations \( nPr \) is given by:

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

And the formula for combinations \( nCr \) is given by:

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

### Step 2: Calculate \( 4P2 \).

Using the permutation formula:

\[ 4P2 = \frac{4!}{(4-2)!} = \frac{4!}{2!} \]

Calculating \( 4! \):

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

So,

\[ 4P2 = \frac{24}{2!} = \frac{24}{2} = 12 \]

### Step 3: Calculate \( 4C2 \).

Using the combination formula:

\[ 4C2 = \frac{4!}{2!(4-2)!} = \frac{4!}{2! \cdot 2!} \]

We already know \( 4! = 24 \) and \( 2! = 2 \):

\[ 4C2 = \frac{24}{2 \cdot 2} = \frac{24}{4} = 6 \]

### Step 4: Substitute the values into the equation.

Now we have:

\[ 12 = n \cdot 6 \]

### Step 5: Solve for \( n \).

To find \( n \), we divide both sides by 6:

\[ n = \frac{12}{6} = 2 \]

### Conclusion

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

\[ \boxed{2} \]