The ratio of coefficients `x^(3) and x^(4)` in the expansion of `(1+x)^(12)` is

Options

A. 4 : 9
B. 1 : 3
C. 2: 3
D. 1: 9

Answer

To find the ratio of the coefficients of \(x^3\) and \(x^4\) in the expansion of \((1+x)^{12}\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Binomial Expansion**:
The binomial expansion of \((1+x)^n\) is given by:
\[ T_{r+1} = \binom{n}{r} x^r \]
where \(T_{r+1}\) is the \((r+1)\)-th term, \(n\) is the exponent, and \(r\) is the term index starting from 0.

2. **Identify the Coefficients**:
For our case, \(n = 12\). We need to find the coefficients of \(x^3\) and \(x^4\).
- The coefficient of \(x^3\) corresponds to \(r = 3\):
\[ \text{Coefficient of } x^3 = \binom{12}{3} \]
- The coefficient of \(x^4\) corresponds to \(r = 4\):
\[ \text{Coefficient of } x^4 = \binom{12}{4} \]

3. **Calculate the Coefficients**:
- Calculate \(\binom{12}{3}\):
\[ \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12!}{3! \cdot 9!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]
- Calculate \(\binom{12}{4}\):
\[ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]

4. **Find the Ratio of the Coefficients**:
Now, we can find the ratio of the coefficients of \(x^3\) and \(x^4\):
\[ \text{Ratio} = \frac{\text{Coefficient of } x^3}{\text{Coefficient of } x^4} = \frac{\binom{12}{3}}{\binom{12}{4}} = \frac{220}{495} \]

5. **Simplify the Ratio**:
To simplify \(\frac{220}{495}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 55:
\[ \frac{220 \div 55}{495 \div 55} = \frac{4}{9} \]

### Final Answer:
The ratio of the coefficients of \(x^3\) and \(x^4\) in the expansion of \((1+x)^{12}\) is \(\frac{4}{9}\).

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