Statement-I : In the expansion of `(1+ x)^n` ifcoefficient of `31^(st) and 32^(nd)` terms are equal then n = 61 Statement -II : Middle term in the expansion of `(1+x)^n` has greatest coefficient.

Answer

To solve the problem step by step, we will analyze both statements provided in the question.

**Step 1: Analyze Statement I**
We need to determine if the coefficients of the 31st and 32nd terms in the expansion of \( (1 + x)^n \) are equal, and if this implies that \( n = 61 \).

The general term \( T_{r+1} \) in the expansion of \( (1 + x)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} x^r \]
where \( \binom{n}{r} \) is the binomial coefficient.

For the 31st term, \( T_{31} \) corresponds to \( r = 30 \):
\[ T_{31} = \binom{n}{30} x^{30} \]
For the 32nd term, \( T_{32} \) corresponds to \( r = 31 \):
\[ T_{32} = \binom{n}{31} x^{31} \]

According to the problem, we set the coefficients equal:
\[ \binom{n}{30} = \binom{n}{31} \]

**Step 2: Use the Property of Binomial Coefficients**
Using the property of binomial coefficients, we know:
\[ \binom{n}{r} = \binom{n}{n-r} \]
Thus, we can express \( \binom{n}{31} \) as:
\[ \binom{n}{31} = \frac{n!}{31!(n-31)!} \]
and \( \binom{n}{30} \) as:
\[ \binom{n}{30} = \frac{n!}{30!(n-30)!} \]

Setting these equal gives:
\[ \frac{n!}{30!(n-30)!} = \frac{n!}{31!(n-31)!} \]

**Step 3: Simplify the Equation**
Cancelling \( n! \) from both sides:
\[ \frac{1}{30!(n-30)!} = \frac{1}{31!(n-31)!} \]
Cross-multiplying yields:
\[ 31!(n-31)! = 30!(n-30)! \]

**Step 4: Further Simplification**
We can express \( 31! \) as \( 31 \times 30! \):
\[ 31 \times 30!(n-31)! = 30!(n-30)! \]
Cancelling \( 30! \) gives:
\[ 31(n-31)! = (n-30)! \]

**Step 5: Expand the Factorials**
We can expand \( (n-30)! \) as:
\[ (n-30)! = (n-30)(n-31)! \]
Substituting this back into the equation gives:
\[ 31(n-31)! = (n-30)(n-31)! \]
Dividing both sides by \( (n-31)! \) (assuming \( n \neq 31 \)):
\[ 31 = n - 30 \]

**Step 6: Solve for \( n \)**
Thus, we find:
\[ n = 61 \]

**Conclusion for Statement I:**
Statement I is true since we have shown that if the coefficients of the 31st and 32nd terms are equal, then \( n = 61 \).

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**Step 7: Analyze Statement II**
The second statement claims that the middle term in the expansion of \( (1 + x)^n \) has the greatest coefficient.

**Step 8: Identify the Middle Term**
For \( n \) even, the middle term is:
\[ T_{(n/2) + 1} = \binom{n}{n/2} x^{n/2} \]
For \( n \) odd, the middle terms are:
\[ T_{(n+1)/2} \text{ and } T_{(n+3)/2} \]

**Step 9: Coefficient Comparison**
The coefficients \( \binom{n}{k} \) increase until \( k = n/2 \) and then decrease. Thus, the middle term (or terms) indeed has the greatest coefficient.

**Conclusion for Statement II:**
Statement II is also true.

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**Final Conclusion:**
Both statements are true.

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