Consider the following statements .
I. The coefficient of the middle term in th expansion of `( 1+ x)^(8)` is equal to the middle term of `( x + ( 1)/( x))^(8)`
II. The coefficient of the middle term in the expansion of `( 1+ x)^(8)` is less than the coefficient of the fifth term in the expansion of `( 1+ x)^(7)` ?
What of the above statements is `//` are correct ?

Options

A. I only
B. II only
C. Both I and II
D. Neither I nor II

Answer

To solve the problem, we need to analyze the two statements given and determine their correctness based on the binomial theorem.

### Step 1: Find the middle term in the expansion of \((1 + x)^8\)

The formula for the \(r\)-th term in the expansion of \((a + b)^n\) is given by:

\[ T_{r} = \binom{n}{r-1} a^{n-(r-1)} b^{(r-1)} \]

In the case of \((1 + x)^8\), the middle term occurs when \(r = \frac{8}{2} + 1 = 5\) (since \(n\) is even).

Thus, the middle term is:

\[ T_{5} = \binom{8}{4} (1)^{8-4} (x)^{4} = \binom{8}{4} x^4 \]

Calculating \(\binom{8}{4}\):

\[ \binom{8}{4} = \frac{8!}{4!4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

So, the middle term in the expansion of \((1 + x)^8\) is \(70x^4\).

### Step 2: Find the middle term in the expansion of \((x + \frac{1}{x})^8\)

For the expression \((x + \frac{1}{x})^8\), we again find the middle term when \(r = 5\):

\[ T_{5} = \binom{8}{4} x^{8-4} \left(\frac{1}{x}\right)^{4} = \binom{8}{4} x^{4} \cdot \frac{1}{x^4} = \binom{8}{4} \]

Calculating \(\binom{8}{4}\) again gives us:

\[ \binom{8}{4} = 70 \]

Thus, the middle term in the expansion of \((x + \frac{1}{x})^8\) is also \(70\).

### Conclusion for Statement I

Both middle terms are equal to \(70\). Therefore, **Statement I is correct**.

### Step 3: Find the fifth term in the expansion of \((1 + x)^7\)

For the expression \((1 + x)^7\), we find the fifth term where \(r = 5\):

\[ T_{5} = \binom{7}{4} (1)^{7-4} (x)^{4} = \binom{7}{4} x^4 \]

Calculating \(\binom{7}{4}\):

\[ \binom{7}{4} = \frac{7!}{4!3!} = \frac{7 \times 6}{2 \times 1} = 21 \]

Thus, the fifth term in the expansion of \((1 + x)^7\) is \(21x^4\).

### Conclusion for Statement II

We compare the coefficients:
- The coefficient of the middle term in \((1 + x)^8\) is \(70\).
- The coefficient of the fifth term in \((1 + x)^7\) is \(21\).

Since \(70\) is not less than \(21\), **Statement II is incorrect**.

### Final Answer

- Statement I is correct.
- Statement II is incorrect.