`""^(7) P_3 =n ""^(7)C_3 ` find the value of n .

Options

A. 4
B. 8
C. 6
D. 9

Answer

To solve the equation \( 7P3 = n \times 7C3 \), we will start by using the formulas for permutations and combinations.

### Step 1: Write the formulas for permutations and combinations.
The formula for permutations is:
\[ nPr = \frac{n!}{(n-r)!} \]
The formula for combinations is:
\[ nCr = \frac{n!}{r!(n-r)!} \]

### Step 2: Substitute the values into the formulas.
For our case, we have:
\[ 7P3 = \frac{7!}{(7-3)!} = \frac{7!}{4!} \]
And for combinations:
\[ 7C3 = \frac{7!}{3! \cdot (7-3)!} = \frac{7!}{3! \cdot 4!} \]

### Step 3: Substitute these into the equation.
Now, substituting these into the original equation:
\[ \frac{7!}{4!} = n \times \frac{7!}{3! \cdot 4!} \]

### Step 4: Simplify the equation.
We can cancel \( 7! \) and \( 4! \) from both sides:
\[ 1 = n \times \frac{1}{3!} \]

### Step 5: Solve for \( n \).
Now, multiplying both sides by \( 3! \):
\[ n = 3! \]
Calculating \( 3! \):
\[ 3! = 3 \times 2 \times 1 = 6 \]

### Final Answer:
Thus, the value of \( n \) is \( 6 \).