In the expansion `(1+x)^7`, the ratio of the coefficients of `x^3` and `x^4` is:

Options

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

Answer

To find the ratio of the coefficients of \(x^3\) and \(x^4\) in the expansion of \((1+x)^7\), we can use the binomial theorem. According to the binomial theorem, the expansion of \((a + b)^n\) is given by:

\[ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

In our case, \(a = 1\), \(b = x\), and \(n = 7\). Therefore, the expansion of \((1+x)^7\) is:

\[ (1+x)^7 = \sum_{k=0}^{7} \binom{7}{k} 1^{7-k} x^k = \sum_{k=0}^{7} \binom{7}{k} x^k \]

### Step 1: Find the coefficient of \(x^3\)

The coefficient of \(x^3\) in the expansion is given by \(\binom{7}{3}\).

\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

### Step 2: Find the coefficient of \(x^4\)

The coefficient of \(x^4\) in the expansion is given by \(\binom{7}{4}\).

\[ \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 35 \]

### Step 3: Calculate the ratio of the coefficients

Now, we need to find the ratio of the coefficients of \(x^3\) and \(x^4\):

\[ \text{Ratio} = \frac{\text{Coefficient of } x^3}{\text{Coefficient of } x^4} = \frac{\binom{7}{3}}{\binom{7}{4}} = \frac{35}{35} = 1 \]

### Final Answer

The ratio of the coefficients of \(x^3\) and \(x^4\) in the expansion of \((1+x)^7\) is \(1\).

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