Understanding Binomial Expansion and Term Coefficients

The problem asks us to find the value of 'r' in the binomial expansion of \(\left(1 + x\right)^{43}\). Specifically, we are given that the coefficients of the \(\left(2r + 1\right)\text{th}\) term and the \(\left(r + 2\right)\text{th}\) term are equal, and that \(r \ne 1\).

The general term in the expansion of \(\left(1 + x\right)^n\) is given by \(T_{k+1} = \binom{n}{k} x^k\). The coefficient of the \(\left(k+1\right)\text{th}\) term is \(\binom{n}{k}\).

In our case, the expansion is \(\left(1 + x\right)^{43}\), so $n=43$.

Identifying the Coefficients

For the \(\left(2r + 1\right)\text{th}\) term, the term number is $2r + 1$. Comparing this to $k+1$, we have $k+1 = 2r + 1$, which means $k = 2r$. The coefficient of the \(\left(2r + 1\right)\text{th}\) term is \(\binom{43}{2r}\).

For the \(\left(r + 2\right)\text{th}\) term, the term number is $r + 2$. Comparing this to $k+1$, we have $k+1 = r + 2$, which means $k = r + 1$. The coefficient of the \(\left(r + 2\right)\text{th}\) term is \(\binom{43}{r+1}\).

Setting the Coefficients Equal

According to the problem statement, the coefficients of the \(\left(2r + 1\right)\text{th}\) and \(\left(r + 2\right)\text{th}\) terms are equal. Therefore, we have:

\(\binom{43}{2r} = \binom{43}{r+1}\)

Applying Binomial Coefficient Properties

We know a property of binomial coefficients that states if \(\binom{n}{a} = \binom{n}{b}\), then either $a = b$ or $a + b = n$.

In our equation, we have $n=43$, $a=2r$, and $b=r+1$. We will consider both cases:

Case 1: $a = b$

Set the lower indices equal:

$2r = r + 1$

Subtract 'r' from both sides:

$2r - r = 1$

$r = 1$

However, the problem statement explicitly says that \(r \ne 1\). Therefore, this case does not give us the required value of r.

Case 2: $a + b = n$

Set the sum of the lower indices equal to the upper index:

$2r + (r + 1) = 43$

Combine the terms with 'r':

$3r + 1 = 43$

Subtract 1 from both sides:

$3r = 43 - 1$

$3r = 42$

Divide by 3:

\(r = \frac{42}{3}\)

$r = 14$

Verifying the Value of r

The value $r=14$ satisfies the condition \(r \ne 1\). Also, the indices $2r$ and $r+1$ must be valid indices for the binomial coefficient \(\binom{43}{k}\), meaning they must be integers between 0 and 43 inclusive.

• For $r=14$, $2r = 2(14) = 28$. Since \(0 \le 28 \le 43\), this index is valid.
• For $r=14$, $r+1 = 14+1 = 15$. Since \(0 \le 15 \le 43\), this index is valid.

Thus, $r=14$ is the valid solution.

The problem asks for the value of r that satisfies the given condition and \(r \ne 1\). Our calculation yields $r=14$, which matches one of the provided options.

Given \(\binom{43}{2r} = \binom{43}{r+1}\), we use the property \(\binom{n}{a} = \binom{n}{b} \implies a=b\) or $a+b=n$.

• \(2r = r+1 \implies r=1\) (Rejected as \(r \ne 1\))
• \(2r + (r+1) = 43 \implies 3r + 1 = 43 \implies 3r = 42 \implies r = 14\) (Accepted)

The value of r is 14.

Revision Table: Binomial Expansion Terms

Additional Information: Combinations and their Properties

The binomial coefficient \(\binom{n}{k}\) is also known as a combination, representing the number of ways to choose $k$ items from a set of $n$ distinct items without regard to the order of selection. It is calculated using the formula:

\(\binom{n}{k} = C(n, k) = \frac{n!}{k!(n-k)!}\)

where $n!$ (n factorial) is the product of all positive integers up to n (\(n! = n \times (n-1) \times \dots \times 2 \times 1\)), and $0! = 1$.

Key properties of binomial coefficients include:

• Symmetry: \(\binom{n}{k} = \binom{n}{n-k}\). This property is directly related to the second case ($a+b=n$) used in the solution. If the coefficient of the \(\left(k+1\right)\text{th}\) term, \(\binom{n}{k}\), is equal to the coefficient of another term, \(\binom{n}{j}\), then either $k=j$ or $k+j=n$. The indices $k$ and $j$ correspond to the 'k' value in the \(T_{k+1}\) formula. In our problem, the terms are the \(\left(2r+1\right)\text{th}\) and \(\left(r+2\right)\text{th}\) terms, corresponding to $k$ values $2r$ and $r+1$ respectively. So, we set \(\binom{43}{2r} = \binom{43}{r+1}\), leading to $2r = r+1$ or $2r + (r+1) = 43$.
• Summation: \(\sum_{k=0}^n \binom{n}{k} = 2^n\). The sum of all coefficients in the expansion of \((1+x)^n\) is \(2^n\).
• Pascal's Identity: \(\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}\). This identity is the basis for constructing Pascal's triangle, where each number is the sum of the two numbers directly above it.

Understanding these properties is crucial for solving problems involving binomial expansion coefficients.