If the coefficients of three consecutive terms inthe expansion of `(1+x)^(n) ` are in the radio `1:5:20`, then the coefficients of the fourth term is.

Options

A. `2436`
B. `3654`
C. `1827`
D. `5481`

Answer

To solve the problem, we need to find the coefficients of three consecutive terms in the expansion of \((1+x)^n\) that are in the ratio \(1:5:20\), and then determine the coefficient of the fourth term.

### Step-by-step Solution:

1. **Identify the Coefficients**:
The coefficients of the terms in the expansion of \((1+x)^n\) are given by the binomial coefficients \(\binom{n}{r}\), \(\binom{n}{r+1}\), and \(\binom{n}{r+2}\) for three consecutive terms.

2. **Set Up the Ratio**:
According to the problem, we have:
\[ \binom{n}{r} : \binom{n}{r+1} : \binom{n}{r+2} = 1 : 5 : 20 \]
This can be expressed as:
\[ \frac{\binom{n}{r+1}}{\binom{n}{r}} = 5 \quad \text{and} \quad \frac{\binom{n}{r+2}}{\binom{n}{r+1}} = 4 \]

3. **Use the Ratio of Binomial Coefficients**:
Using the property of binomial coefficients:
\[ \frac{\binom{n}{r+1}}{\binom{n}{r}} = \frac{n-r}{r+1} = 5 \]
This gives us the equation:
\[ n - r = 5(r + 1) \implies n - r = 5r + 5 \implies n = 6r + 5 \quad \text{(1)} \]

4. **Set Up the Second Ratio**:
For the second ratio:
\[ \frac{\binom{n}{r+2}}{\binom{n}{r+1}} = \frac{n - (r + 1)}{r + 2} = 4 \]
This gives us the equation:
\[ n - (r + 1) = 4(r + 2) \implies n - r - 1 = 4r + 8 \implies n = 5r + 9 \quad \text{(2)} \]

5. **Solve the System of Equations**:
Now we have two equations:
\[ n = 6r + 5 \quad \text{(1)} \]
\[ n = 5r + 9 \quad \text{(2)} \]
Setting these equal to each other:
\[ 6r + 5 = 5r + 9 \]
Solving for \(r\):
\[ 6r - 5r = 9 - 5 \implies r = 4 \]

6. **Substituting \(r\) Back to Find \(n\)**:
Substitute \(r = 4\) into either equation to find \(n\):
\[ n = 6(4) + 5 = 24 + 5 = 29 \]

7. **Finding the Coefficient of the Fourth Term**:
The coefficient of the fourth term in the expansion of \((1+x)^n\) is given by:
\[ \text{Coefficient of } x^3 = \binom{n}{3} = \binom{29}{3} \]
Calculating \(\binom{29}{3}\):
\[ \binom{29}{3} = \frac{29 \times 28 \times 27}{3 \times 2 \times 1} = \frac{21952}{6} = 3654 \]

### Final Answer:
The coefficient of the fourth term is **3654**.