If `(""^(28)C_(2r): ""^(24)C_(2r-4))=225:11 .`Find the value of r.

Options

A. 10
B. 11
C. 7
D. 9

Answer

To solve the problem, we need to find the value of \( r \) given the equation:

\[ \frac{^{28}C_{2r}}{^{24}C_{2r-4}} = \frac{225}{11} \]

### Step 1: Write the combinations in terms of factorials
The combination formula is given by:

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

Using this, we can express \( ^{28}C_{2r} \) and \( ^{24}C_{2r-4} \):

\[ ^{28}C_{2r} = \frac{28!}{(2r)!(28-2r)!} \]

\[ ^{24}C_{2r-4} = \frac{24!}{(2r-4)!(24-(2r-4))!} = \frac{24!}{(2r-4)!(28-2r)!} \]

### Step 2: Substitute the combinations into the equation
Now substituting these into the ratio gives:

\[ \frac{\frac{28!}{(2r)!(28-2r)!}}{\frac{24!}{(2r-4)!(28-2r)!}} = \frac{225}{11} \]

### Step 3: Simplify the equation
This simplifies to:

\[ \frac{28!}{(2r)!(28-2r)!} \cdot \frac{(2r-4)!(28-2r)!}{24!} = \frac{225}{11} \]

The \( (28-2r)! \) cancels out:

\[ \frac{28! \cdot (2r-4)!}{(2r)! \cdot 24!} = \frac{225}{11} \]

### Step 4: Further simplify the left-hand side
We can express \( 28! \) as \( 28 \times 27 \times 26 \times 25 \times 24! \):

\[ \frac{28 \times 27 \times 26 \times 25 \times 24! \cdot (2r-4)!}{(2r)! \cdot 24!} = \frac{225}{11} \]

This simplifies to:

\[ \frac{28 \times 27 \times 26 \times 25 \cdot (2r-4)!}{(2r)!} = \frac{225}{11} \]

### Step 5: Express \( (2r)! \) in terms of \( (2r-4)! \)
We can express \( (2r)! \) as:

\[ (2r)! = (2r)(2r-1)(2r-2)(2r-3)(2r-4)! \]

Substituting this back gives:

\[ \frac{28 \times 27 \times 26 \times 25}{(2r)(2r-1)(2r-2)(2r-3)} = \frac{225}{11} \]

### Step 6: Cross-multiply to solve for \( r \)
Cross-multiplying gives:

\[ 11 \cdot (28 \times 27 \times 26 \times 25) = 225 \cdot (2r)(2r-1)(2r-2)(2r-3) \]

### Step 7: Calculate the left-hand side
Calculating \( 28 \times 27 \times 26 \times 25 \):

\[ 28 \times 27 = 756, \quad 756 \times 26 = 19656, \quad 19656 \times 25 = 491400 \]

Thus,

\[ 11 \cdot 491400 = 5405400 \]

### Step 8: Set up the equation
Now we have:

\[ 5405400 = 225 \cdot (2r)(2r-1)(2r-2)(2r-3) \]

### Step 9: Divide both sides by 225
Calculating \( \frac{5405400}{225} \):

\[ \frac{5405400}{225} = 24024 \]

So we have:

\[ (2r)(2r-1)(2r-2)(2r-3) = 24024 \]

### Step 10: Test possible values for \( r \)
We can test the options given (10, 11, 7, 9) for \( r \):

1. For \( r = 7 \):
\[ (2 \cdot 7)(2 \cdot 7 - 1)(2 \cdot 7 - 2)(2 \cdot 7 - 3) = 14 \cdot 13 \cdot 12 \cdot 11 \]
Calculating:
\[ 14 \cdot 13 = 182, \quad 182 \cdot 12 = 2184, \quad 2184 \cdot 11 = 24024 \]

Thus, \( r = 7 \) satisfies the equation.

### Final Answer
The value of \( r \) is \( 7 \).