If `""^(10)P_(r)` = 720, find r.

Options

A. a) 2
B. b) 3
C. c) 5
D. d ) 4

Answer

To solve the problem where \( 10P_r = 720 \), we will use the formula for permutations. The formula for permutations is given by:

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

In our case, \( n = 10 \) and we need to find \( r \) such that:

\[ 10P_r = \frac{10!}{(10-r)!} = 720 \]

### Step 1: Set up the equation
We can set up the equation based on the permutation formula:

\[ \frac{10!}{(10-r)!} = 720 \]

### Step 2: Calculate \( 10! \)
We know that \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \). However, for our purposes, we can simplify our calculations by not calculating the entire factorial, but rather focusing on the terms we need.

### Step 3: Rewrite the equation
We can rewrite the equation as:

\[ 10 \times 9 \times 8 \times \ldots \times (10 - r + 1) = 720 \]

This means we will multiply the first \( r \) terms starting from 10 downwards.

### Step 4: Test values for \( r \)
Now, we will test different values of \( r \) to find which one satisfies the equation.

- **For \( r = 2 \)**:
\[ 10P_2 = 10 \times 9 = 90 \quad (\text{not } 720) \]

- **For \( r = 3 \)**:
\[ 10P_3 = 10 \times 9 \times 8 = 720 \quad (\text{satisfies the equation}) \]

### Step 5: Conclusion
Since \( 10P_3 = 720 \), we find that:

\[ r = 3 \]

Thus, the value of \( r \) is \( 3 \).