The greatest term in the expansion of `(3 + 5x)^(15)`, when x=1/5, is

Options

A. `""^(15)C_(3)(3^(13))`
B. `""^(15)C_(4)(3^(12))`
C. `""^(15)C_(4)(3^(10))`
D. none of these

Answer

To find the greatest term in the expansion of \((3 + 5x)^{15}\) when \(x = \frac{1}{5}\), we will follow these steps:

### Step 1: Identify the General Term
The general term \(T_{r+1}\) in the expansion of \((a + b)^n\) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
For our case, \(a = 3\), \(b = 5x\), and \(n = 15\). Thus, the general term becomes:
\[ T_{r+1} = \binom{15}{r} (3)^{15-r} (5x)^r \]

### Step 2: Substitute \(x = \frac{1}{5}\)
Substituting \(x = \frac{1}{5}\) into the general term:
\[ T_{r+1} = \binom{15}{r} (3)^{15-r} (5 \cdot \frac{1}{5})^r = \binom{15}{r} (3)^{15-r} (1)^r = \binom{15}{r} (3)^{15-r} \]

### Step 3: Find the Value of \(r\) for the Greatest Term
To find the greatest term, we need to determine the value of \(r\) that maximizes \(T_{r+1}\). The greatest term occurs at:
\[ r = \left\lfloor \frac{n \cdot b}{a + b} \right\rfloor \quad \text{or} \quad r = \left\lfloor \frac{n \cdot 5}{3 + 5} \right\rfloor \]
Calculating this:
\[ r = \left\lfloor \frac{15 \cdot 5}{8} \right\rfloor = \left\lfloor \frac{75}{8} \right\rfloor = \left\lfloor 9.375 \right\rfloor = 9 \]

### Step 4: Calculate the Greatest Term
Now, we will calculate \(T_{10}\) (since \(r = 9\)):
\[ T_{10} = \binom{15}{9} (3)^{15-9} = \binom{15}{9} (3)^6 \]

### Step 5: Calculate \(\binom{15}{9}\)
Using the property of binomial coefficients:
\[ \binom{15}{9} = \binom{15}{6} = \frac{15!}{6! \cdot 9!} \]
Calculating this:
\[ \binom{15}{6} = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 5005 \]

### Step 6: Calculate \(3^6\)
Calculating \(3^6\):
\[ 3^6 = 729 \]

### Step 7: Final Calculation of the Greatest Term
Now we can find the greatest term:
\[ T_{10} = 5005 \times 729 \]

Calculating this:
\[ T_{10} = 3643650 \]

### Conclusion
Thus, the greatest term in the expansion of \((3 + 5x)^{15}\) when \(x = \frac{1}{5}\) is:
\[ \boxed{3643650} \]