Key Concept: The Binomial Theorem and General Term

The Binomial Theorem provides a formula for expanding any power of a binomial $(a+b)^n$. A crucial part of this theorem is the general term, often denoted as $T_{r+1}$, which allows us to find any specific term in the expansion without writing out the entire series. The formula for the $(r+1)^{th}$ term in the expansion of $(a+b)^n$ is given by:

$T_{r+1} = {}^nC_r a^{n-r} b^r$

where ${}^nC_r = \frac{n!}{r!(n-r)!}$ is the binomial coefficient. This formula is fundamental for problems asking for specific coefficients of powers of $x$.

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Step-by-Step Solution

Part 1: Finding the coefficient of $x^7$ in ${\left( {{x^2} + {1 \over {bx}}} \right)^{11}}$

1. Identify $a$, $b$, and $n$: For the expression ${\left( {{x^2} + {1 \over {bx}}} \right)^{11}}$, we have:

   • $a = x^2$
   • $b = \frac{1}{bx}$
   • $n = 11$

2. Write the General Term ($T_{r+1}$): Using the general term formula $T_{r+1} = {}^nC_r a^{n-r} b^r$: $T_{r+1} = {}^{11}C_r {(x^2)}^{11-r} {\left( {\frac{1}{bx}} \right)^r}$

   • Explanation: We substitute the specific values of $a$, $b$, and $n$ into the general term formula to get an expression for any term in the expansion.

3. Simplify the General Term to collect powers of $x$: $T_{r+1} = {}^{11}C_r x^{2(11-r)} \cdot \frac{1}{b^r x^r}$ $T_{r+1} = {}^{11}C_r x^{22-2r} \cdot b^{-r} \cdot x^{-r}$ $T_{r+1} = {}^{11}C_r b^{-r} x^{22-2r-r}$ $T_{r+1} = {}^{11}C_r b^{-r} x^{22-3r}$

   • Explanation: The goal here is to isolate the powers of $x$ so we can equate the exponent to the desired power. We use exponent rules $(x^m)^n = x^{mn}$ and $\frac{1}{x^m} = x^{-m}$ to simplify.

4. Find the value of $r$ for the term containing $x^7$: We need the coefficient of $x^7$, so we equate the exponent of $x$ in the simplified general term to 7: $22 - 3r = 7$ $3r = 22 - 7$ $3r = 15$ $r = 5$

   • Explanation: Solving for $r$ tells us which specific term (the $(r+1)^{th}$ term) in the binomial expansion contains the desired power of $x$. Since $r$ must be a non-negative integer, this indicates a valid term exists.

5. Determine the coefficient of $x^7$: Substitute $r=5$ back into the coefficient part of the general term ($^{11}C_r b^{-r}$): $\text{Coefficient of } x^7 = {}^{11}C_5 b^{-5} = \frac{{}^{11}C_5}{b^5}$

   • Explanation: Once $r$ is found, we plug it back into the part of the general term that does not include $x$ to get the numerical coefficient.

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Part 2: Finding the coefficient of $x^{-7}$ in ${\left( {x - {1 \over {bx^2}}} \right)^{11}}$

1. Identify $a$, $b$, and $n$: For the expression ${\left( {x - {1 \over {bx^2}}} \right)^{11}}$, we have:

   • $a = x$
   • $b = -\frac{1}{bx^2}$ (Note the negative sign!)
   • $n = 11$

2. Write the General Term ($T_{k+1}$): Let's use $k$ instead of $r$ for the term number here to avoid confusion between the two expansions. $T_{k+1} = {}^{11}C_k {(x)}^{11-k} {\left( { - \frac{1}{bx^2}} \right)^k}$

   • Explanation: Similar to Part 1, we apply the general term formula. It's crucial to include the negative sign with the second term, as $b$ in $(a+b)^n$ includes its sign.

3. Simplify the General Term to collect powers of $x$: $T_{k+1} = {}^{11}C_k x^{11-k} \cdot {(-1)^k} \cdot \frac{1}{b^k x^{2k}}$ $T_{k+1} = {}^{11}C_k {(-1)^k} b^{-k} x^{11-k-2k}$ $T_{k+1} = {}^{11}C_k {(-1)^k} b^{-k} x^{11-3k}$

   • Explanation: Again, we simplify to isolate powers of $x$. Be careful with the negative sign; $(-1)^k$ will be part of the coefficient.

4. Find the value of $k$ for the term containing $x^{-7}$: We need the coefficient of $x^{-7}$, so we equate the exponent of $x$ to -7: $11 - 3k = -7$ $3k = 11 + 7$ $3k = 18$ $k = 6$

   • Explanation: Solving for $k$ identifies the term containing $x^{-7}$.

5. Determine the coefficient of $x^{-7}$: Substitute $k=6$ back into the coefficient part of the general term ($^{11}C_k (-1)^k b^{-k}$): $\text{Coefficient of } x^{-7} = {}^{11}C_6 {(-1)^6} b^{-6}$ Since ${(-1)^6} = 1$: $\text{Coefficient of } x^{-7} = {}^{11}C_6 \cdot 1 \cdot b^{-6} = \frac{{}^{11}C_6}{b^6}$

   • Explanation: Substituting $k=6$ into the coefficient part gives us the required coefficient. The even power of $(-1)$ makes it positive.

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Part 3: Equating the coefficients and solving for $b$

1. Set the two coefficients equal: The problem states that the coefficients are equal: $\frac{{}^{11}C_5}{b^5} = \frac{{}^{11}C_6}{b^6}$

   • Explanation: This is the core condition given by the problem statement, translating the verbal description into an algebraic equation.

2. Solve for $b$: Since $b \ne 0$ (given in the question), we can multiply both sides by $b^6$ to clear the denominators: ${}^{11}C_5 \cdot b = {}^{11}C_6$

   • Explanation: Multiplying by $b^6$ simplifies the equation, allowing us to solve for $b$. If $b$ could be zero, this step would require careful consideration.

   Recall a useful property of binomial coefficients: ${}^nC_r = {}^nC_{n-r}$. Applying this property, we know that ${}^{11}C_5 = {}^{11}C_{11-5} = {}^{11}C_6$.

   • Explanation: This property is a key shortcut. Recognizing it simplifies the calculation significantly. It means that choosing 5 items from 11 is the same as choosing (or leaving behind) 6 items from 11.

   Now, substitute ${}^{11}C_5$ with ${}^{11}C_6$ in our equation: ${}^{11}C_6 \cdot b = {}^{11}C_6$

   Since ${}^{11}C_6$ is a non-zero constant, we can divide both sides by ${}^{11}C_6$: $b = 1$

   • Explanation: Dividing by the non-zero binomial coefficient isolates $b$, giving us its value.

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Tips and Common Mistakes:

• Don't forget the sign: When the second term of the binomial is negative (e.g., $(a-b)^n$), ensure the negative sign is included when applying the general term formula, often leading to a $(-1)^r$ factor.
• Careful with exponents: Pay close attention to combining exponents, especially when dealing with terms like $(x^2)^r$ or $1/x^r$.
• Binomial Coefficient Property: Remember ${}^nC_r = {}^nC_{n-r}$. This can often simplify equations involving binomial coefficients.
• Check conditions: The problem explicitly states $b \ne 0$, which justifies multiplying by $b^6$. Always note such conditions.

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Summary and Key Takeaway:

This problem effectively tests your understanding of the Binomial Theorem's general term and your ability to algebraically manipulate exponents and binomial coefficients. The key is to systematically find the general term for each binomial, simplify to isolate the power of $x$, solve for the term number ($r$ or $k$), and then extract the coefficients. Recognizing properties like ${}^nC_r = {}^nC_{n-r}$ can significantly streamline the final steps of solving the equation. The value of $b$ is $\boxed{1}$.