If the coefficient of the `5^(th)` term be the numerically the greatest coefficient in the expansion of `(1-x)^n`, then the positiveintegral value of n is

Options

A. 9
B. 8
C. 7
D. 10

Explanation

To solve the problem, we need to find the positive integral value of \( n \) such that the coefficient of the 5th term in the expansion of \( (1 - x)^n \) is the numerically greatest coefficient.

### Step-by-Step Solution:

1. **Understanding the Expansion**:
The binomial expansion of \( (1 - x)^n \) can be expressed as:
\[ (1 - x)^n = \sum_{k=0}^{n} \binom{n}{k} (-x)^k \]
The \( k^{th} \) term in this expansion is given by:
\[ T_{k+1} = \binom{n}{k} (-x)^k \]

2. **Identifying the 5th Term**:
The 5th term corresponds to \( k = 4 \) (since the first term corresponds to \( k = 0 \)):
\[ T_5 = \binom{n}{4} (-x)^4 = \binom{n}{4} x^4 \]

3. **Finding the Middle Term**:
The total number of terms in the expansion is \( n + 1 \). The middle term occurs when:
\[ \frac{n + 1}{2} \]
For the 5th term to be the middle term, we set:
\[ \frac{n + 1}{2} = 5 \]

4. **Solving for \( n \)**:
Multiply both sides by 2:
\[ n + 1 = 10 \]
Subtract 1 from both sides:
\[ n = 9 \]

5. **Conclusion**:
The positive integral value of \( n \) is \( 9 \).

### Final Answer:
The positive integral value of \( n \) is \( 9 \).

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