Detailed Solution for Binomial Expansion Coefficient Sum

1. Key Concept: The General Term of a Binomial Expansion

The binomial theorem states that for any real numbers $a$ and $b$, and any non-negative integer $n$, the expansion of $(a+b)^n$ is given by: $(a+b)^n = \sum_{r=0}^n {}^n C_r a^{n-r} b^r$ The $(r+1)^{th}$ term, often denoted as $T_{r+1}$, in this expansion is: $T_{r+1} = {}^n C_r a^{n-r} b^r$ This general term allows us to find any specific term or analyze the powers of variables within the expansion.

2. Applying the General Term to the Given Expression

We are given the binomial expansion of ${\left( {2{x^3} + {3 \over x}} \right)^{10}}$. Here, we identify:

• $a = 2x^3$
• $b = \frac{3}{x} = 3x^{-1}$
• $n = 10$

Substituting these into the general term formula: $T_{r+1} = {}^{10}{C_r} \cdot (2{x^3})^{10-r} \cdot \left(\frac{3}{x}\right)^r$ Now, we simplify this expression to separate the numerical coefficient part from the variable part (powers of $x$): $T_{r+1} = {}^{10}{C_r} \cdot 2^{10-r} \cdot (x^3)^{10-r} \cdot 3^r \cdot (x^{-1})^r$ Using exponent rules $(x^m)^n = x^{mn}$ and $x^m \cdot x^n = x^{m+n}$: $T_{r+1} = {}^{10}{C_r} \cdot 2^{10-r} \cdot x^{3(10-r)} \cdot 3^r \cdot x^{-r}$ $T_{r+1} = {}^{10}{C_r} \cdot 2^{10-r} \cdot 3^r \cdot x^{30-3r-r}$ $T_{r+1} = {}^{10}{C_r} \cdot 2^{10-r} \cdot 3^r \cdot x^{30-4r}$ The coefficient of $x^{30-4r}$ in the expansion is $C_r = {}^{10}{C_r} \cdot 2^{10-r} \cdot 3^r$.

3. Determining Conditions for Positive Even Powers of x

We are interested in the sum of coefficients of all positive even powers of $x$. This means two conditions must be met for the exponent $(30-4r)$:

1. Positive: The exponent must be greater than zero, i.e., $30-4r > 0$.
2. Even: The exponent must be an even integer.

Let's analyze the exponent $30-4r$:

• Since $30$ is an even number and $4r$ (where $r$ is an integer) is always an even number, their difference $(30-4r)$ will always be an even number. Therefore, the "even" condition is automatically satisfied for all integer values of $r$.
• Now, let's consider the "positive" condition: $30 - 4r > 0$ $30 > 4r$ $r < \frac{30}{4}$ $r < 7.5$
• Since $r$ can only take integer values from $0$ to $n=10$ in the binomial expansion, the possible values of $r$ that satisfy $r < 7.5$ are $r \in \{0, 1, 2, 3, 4, 5, 6, 7\}$.

Let's quickly check the powers of $x$ for these values of $r$:

• For $r=0$: $x^{30-4(0)} = x^{30}$ (positive, even)
• For $r=1$: $x^{30-4(1)} = x^{26}$ (positive, even)
• For $r=2$: $x^{30-4(2)} = x^{22}$ (positive, even)
• For $r=3$: $x^{30-4(3)} = x^{18}$ (positive, even)
• For $r=4$: $x^{30-4(4)} = x^{14}$ (positive, even)
• For $r=5$: $x^{30-4(5)} = x^{10}$ (positive, even)
• For $r=6$: $x^{30-4(6)} = x^6$ (positive, even)
• For $r=7$: $x^{30-4(7)} = x^2$ (positive, even)

These are the terms whose coefficients we need to sum.

4. Formulating the Sum of Coefficients

The sum of the coefficients for these positive even powers of $x$ is: $S = \sum_{r=0}^{7} {}^{10}{C_r} \cdot 2^{10-r} \cdot 3^r$ This sum can be written as: $S = {}^{10}{C_0} \cdot 2^{10} \cdot 3^0 + {}^{10}{C_1} \cdot 2^9 \cdot 3^1 + \dots + {}^{10}{C_7} \cdot 2^3 \cdot 3^7$

5. Utilizing the Full Binomial Expansion Identity

A powerful technique for sums of coefficients is to consider the full binomial expansion. The sum of all coefficients in the expansion of $(a+b)^n$ is $(a+b)^n$ itself if we evaluate it at $x=1$ (or whatever variable is present). However, here, the coefficients are not simple integer constants. The coefficient of each term $T_{r+1}$ is ${}^{10}{C_r} \cdot 2^{10-r} \cdot 3^r$.

Let's consider the full expansion of $(2+3)^{10}$. This is effectively the sum of all coefficients ${}^{10}{C_r} \cdot 2^{10-r} \cdot 3^r$ for $r$ from $0$ to $10$. $(2+3)^{10} = \sum_{r=0}^{10} {}^{10}{C_r} \cdot 2^{10-r} \cdot 3^r = {}^{10}{C_0}2^{10}3^0 + \dots + {}^{10}{C_7}2^33^7 + {}^{10}{C_8}2^23^8 + {}^{10}{C_9}2^13^9 + {}^{10}{C_{10}}2^03^{10}$ The sum we want, $S$, includes terms from $r=0$ to $r=7$. The full sum $(2+3)^{10}$ includes terms for $r=8, 9, 10$ as well. Let's examine the powers of $x$ for these remaining terms:

• For $r=8$: $x^{30-4(8)} = x^{30-32} = x^{-2}$ (even, but not positive)
• For $r=9$: $x^{30-4(9)} = x^{30-36} = x^{-6}$ (even, but not positive)
• For $r=10$: $x^{30-4(10)} = x^{30-40} = x^{-10}$ (even, but not positive)

Since the problem asks only for positive even powers of $x$, we need to subtract the coefficients of these terms (for $r=8, 9, 10$) from the total sum $(2+3)^{10}$.

So, the sum we want is: $S = (2+3)^{10} - \left( {}^{10}{C_8} \cdot 2^2 \cdot 3^8 + {}^{10}{C_9} \cdot 2^1 \cdot 3^9 + {}^{10}{C_{10}} \cdot 2^0 \cdot 3^{10} \right)$ $S = 5^{10} - \left( {}^{10}{C_8} \cdot 4 \cdot 3^8 + {}^{10}{C_9} \cdot 2 \cdot 3^9 + {}^{10}{C_{10}} \cdot 1 \cdot 3^{10} \right)$

6. Calculating the Subtracted Terms

Let's calculate the binomial coefficients:

• ${}^{10}{C_8} = {}^{10}{C_{10-8}} = {}^{10}{C_2} = \frac{10 \times 9}{2 \times 1} = 45$
• ${}^{10}{C_9} = {}^{10}{C_{10-9}} = {}^{10}{C_1} = 10$
• ${}^{10}{C_{10}} = 1$

Substitute these values back into the expression: $S = 5^{10} - \left( 45 \cdot 4 \cdot 3^8 + 10 \cdot 2 \cdot 3^9 + 1 \cdot 1 \cdot 3^{10} \right)$ $S = 5^{10} - \left( 180 \cdot 3^8 + 20 \cdot 3^9 + 3^{10} \right)$ To factor out $3^9$, we need to rewrite $3^8$ and $3^{10}$:

• $3^8 = \frac{3^9}{3}$
• $3^{10} = 3 \cdot 3^9$

Substitute these into the expression: $S = 5^{10} - \left( 180 \cdot \frac{3^9}{3} + 20 \cdot 3^9 + 3 \cdot 3^9 \right)$ $S = 5^{10} - \left( 60 \cdot 3^9 + 20 \cdot 3^9 + 3 \cdot 3^9 \right)$ Now, factor out $3^9$: $S = 5^{10} - (60 + 20 + 3) \cdot 3^9$ $S = 5^{10} - 83 \cdot 3^9$

7. Solving for $\beta$

The problem states that the sum of coefficients is ${5^{10}} - \beta \,.\,{3^9}$. By comparing our result $S = 5^{10} - 83 \cdot 3^9$ with the given form, we can clearly see that: $\beta = 83$

Tips and Common Mistakes:

• Careful with Exponents: A common mistake is to miscalculate the final power of $x$ (e.g., $x^{30-3r-r}$ can easily become $x^{30-3r}$ if not careful).
• Don't Forget the Bounds of r: Remember that $r$ ranges from $0$ to $n$ (here, $0$ to $10$).
• Distinguish Positive from Non-Negative: The problem explicitly asks for positive powers, so $x^0$ (if it were even) would not be included, and negative powers are certainly excluded.
• Factoring for Comparison: When comparing with the target form (like $5^{10} - \beta \cdot 3^9$), make sure to factor out the common term ($3^9$ in this case) correctly.

Summary: We first determined the general term of the binomial expansion and identified the power of $x$ as $30-4r$. By applying the conditions for "positive even powers of x", we found that $r$ can range from $0$ to $7$. We then used the identity for the full binomial sum $(2+3)^{10}$ and subtracted the coefficients of terms whose powers of $x$ were even but not positive (i.e., negative powers, corresponding to $r=8, 9, 10$). After calculating these excluded terms and simplifying, we found the value of $\beta$.