Introduction: Understanding the Problem

The problem asks us to find the constant term in the binomial expansion of ${\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}}$. The constant term is then expressed in the form $2^k \cdot l$, where $l$ is an odd integer, and we need to determine the value of $k$. This type of problem typically involves the Multinomial Theorem to handle expansions with more than two terms.

Key Concept: The Multinomial Theorem

For an expression of the form $(x_1 + x_2 + ... + x_m)^n$, the general term in its expansion is given by: $T = \frac{n!}{n_1! n_2! ... n_m!} x_1^{n_1} x_2^{n_2} ... x_m^{n_m}$ where:

• $n$ is the total power of the expansion.
• $x_1, x_2, ..., x_m$ are the individual terms within the parenthesis.
• $n_1, n_2, ..., n_m$ are non-negative integers representing the powers of each term in a specific product, such that their sum equals $n$: $n_1 + n_2 + ... + n_m = n$ The "constant term" is the term where the power of the variable (in this case, $x$) is zero, i.e., it is the coefficient of $x^0$.

Step 1: Simplify the Expression to Isolate the Variable Part

The given expression is ${\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}}$. To simplify, we can take $x^5$ as a common denominator inside the parenthesis: ${\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}} = {\left( {{{3{x^3} \cdot {x^5}} \over {{x^5}}} - {{2{x^2} \cdot {x^5}} \over {{x^5}}} + {5 \over {{x^5}}}} \right)^{10}}$ $= {\left( {{{3{x^8} - 2{x^7} + 5} \over {{x^5}}}} \right)^{10}}$ Now, we can separate the numerator and denominator: $= {{{{\left( {3{x^8} - 2{x^7} + 5} \right)}^{10}}} \over {{{\left( {{x^5}} \right)}^{10}}}} = {{{{\left( {3{x^8} - 2{x^7} + 5} \right)}^{10}}} \over {{x^{50}}}}$ For this entire expression to be a constant term (i.e., independent of $x$), the numerator must contain an $x^{50}$ term that cancels out the $x^{50}$ in the denominator. Therefore, our goal is to find the coefficient of $x^{50}$ in the expansion of ${\left( {3{x^8} - 2{x^7} + 5} \right)^{10}}$.

Step 2: Apply the Multinomial Theorem to Find the General Term of the Numerator

Let's consider the expansion of ${\left( {3{x^8} - 2{x^7} + 5} \right)^{10}}$. Here, $n=10$, and the terms are:

• $x_1 = 3x^8$
• $x_2 = -2x^7$
• $x_3 = 5$ (which can be considered as $5x^0$)

Using the Multinomial Theorem, the general term is: $T = \frac{10!}{n_1! n_2! n_3!} (3x^8)^{n_1} (-2x^7)^{n_2} (5)^{n_3}$ where $n_1, n_2, n_3$ are non-negative integers such that $n_1 + n_2 + n_3 = 10$.

Let's separate the numerical coefficients and the powers of $x$: $T = \frac{10!}{n_1! n_2! n_3!} (3)^{n_1} (x^8)^{n_1} (-2)^{n_2} (x^7)^{n_2} (5)^{n_3} (x^0)^{n_3}$ $T = \frac{10!}{n_1! n_2! n_3!} (3)^{n_1} (-2)^{n_2} (5)^{n_3} x^{8n_1} x^{7n_2} x^0$ Combining the powers of $x$: $T = \left( \frac{10!}{n_1! n_2! n_3!} (3)^{n_1} (-2)^{n_2} (5)^{n_3} \right) x^{8n_1 + 7n_2}$ The coefficient of $x^{8n_1 + 7n_2}$ is the expression in the parenthesis.

Step 3: Determine Conditions for the Desired Term ($x^{50}$)

For the term to be $x^{50}$, the exponent of $x$ must be 50. So, we set up the equation: $8n_1 + 7n_2 = 50 \quad (*)$ We also have the fundamental constraint from the Multinomial Theorem: $n_1 + n_2 + n_3 = 10 \quad (**)$ Remember that $n_1, n_2, n_3$ must be non-negative integers ($n_i \ge 0$).

Step 4: Find Valid Integer Values for $n_1, n_2, n_3$

We need to find non-negative integer solutions for $n_1$ and $n_2$ for the equation $8n_1 + 7n_2 = 50$. We can systematically test values for $n_1$:

• If $n_1 = 0$: $7n_2 = 50 \implies n_2 = 50/7$, which is not an integer.
• If $n_1 = 1$: $8(1) + 7n_2 = 50 \implies 7n_2 = 42 \implies n_2 = 6$.
  • This is a valid integer solution for $(n_1, n_2)$. Now, find $n_3$ using $n_1 + n_2 + n_3 = 10$:
  • $1 + 6 + n_3 = 10 \implies 7 + n_3 = 10 \implies n_3 = 3$.
  • So, $(n_1, n_2, n_3) = (1, 6, 3)$ is a valid set of indices. All are non-negative integers.
• If $n_1 = 2$: $8(2) + 7n_2 = 50 \implies 16 + 7n_2 = 50 \implies 7n_2 = 34$, not an integer.
• If $n_1 = 3$: $8(3) + 7n_2 = 50 \implies 24 + 7n_2 = 50 \implies 7n_2 = 26$, not an integer.
• If $n_1 = 4$: $8(4) + 7n_2 = 50 \implies 32 + 7n_2 = 50 \implies 7n_2 = 18$, not an integer.
• If $n_1 = 5$: $8(5) + 7n_2 = 50 \implies 40 + 7n_2 = 50 \implies 7n_2 = 10$, not an integer.
• If $n_1 = 6$: $8(6) + 7n_2 = 50 \implies 48 + 7n_2 = 50 \implies 7n_2 = 2$, not an integer.
• If $n_1 = 7$: $8(7) = 56$, which is already greater than 50. So, no further solutions for $n_1 \ge 7$ are possible for non-negative $n_2$.

Thus, the only combination of $(n_1, n_2, n_3)$ that satisfies all conditions is $(1, 6, 3)$.

Step 5: Calculate the Coefficient of $x^{50}$

Now we substitute $n_1=1, n_2=6, n_3=3$ into the coefficient part of the general term: $\text{Coefficient} = \frac{10!}{1! 6! 3!} (3)^{1} (-2)^{6} (5)^{3}$ Let's calculate the factorial part: $\frac{10!}{1! 6! 3!} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{1 \times 6! \times (3 \times 2 \times 1)} = \frac{10 \times 9 \times 8 \times 7}{6}$ $= 10 \times 3 \times 4 \times 7 = 840$ Now substitute this back into the coefficient expression: $\text{Coefficient} = 840 \times (3)^1 \times (-2)^6 \times (5)^3$ $= 840 \times 3 \times 64 \times 125$ Let's perform the multiplication and factorize into prime powers: $840 = 84 \times 10 = (2^2 \times 3 \times 7) \times (2 \times 5) = 2^3 \times 3 \times 5 \times 7$ $3 = 3^1$ $64 = 2^6$ $125 = 5^3$ So, $\text{Coefficient} = (2^3 \times 3 \times 5 \times 7) \times 3^1 \times 2^6 \times 5^3$ Group the powers of each prime number: $\text{Coefficient} = 2^{3+6} \times 3^{1+1} \times 5^{1+3} \times 7^1$ $\text{Coefficient} = 2^9 \times 3^2 \times 5^4 \times 7$

Step 6: Express in the Required Form and Find $k$

The problem states that the constant term is $2^k \cdot l$, where $l$ is an odd integer. Our calculated constant term is $2^9 \times 3^2 \times 5^4 \times 7$. Comparing this with $2^k \cdot l$:

• The power of 2 is $2^9$, so $k=9$.
• The remaining part, $l = 3^2 \times 5^4 \times 7 = 9 \times 625 \times 7 = 5625 \times 7 = 39375$. We check that $l = 39375$ is indeed an odd integer (it is not divisible by 2).

Therefore, the value of $k$ is $9$.

Tip on Common Mistakes:

• Algebraic Error in Simplification: A common mistake is to incorrectly simplify the original expression, as shown in the initial thought process, leading to a different problem. Always double-check algebraic manipulations.
• Incorrectly Identifying Terms: Ensure that each term ($x_1, x_2, x_3$) is correctly identified, including its sign and any powers of $x$.
• Forgetting Constraints: It's crucial to remember that $n_1, n_2, n_3$ must be non-negative integers that sum up to $n$. Systematically testing values helps prevent overlooking solutions or including invalid ones.
• Factorization Errors: When expressing the final coefficient in the form $2^k \cdot l$, ensure careful prime factorization to correctly identify $k$ and confirm $l$ is odd.

Summary and Key Takeaway

To find the constant term in a multinomial expansion, especially with complex terms involving variables in the denominator, it's often helpful to first rewrite the entire expression such that the variable powers are easily combined. Then, apply the Multinomial Theorem to define the general term. The core of the problem lies in setting the total exponent of the variable to zero (for the constant term) or to a specific value (if the expression was simplified) and systematically finding all non-negative integer combinations for the powers of individual terms ($n_i$). Finally, calculate the coefficient for these valid combinations and express it in the desired format. In this case, careful calculation led to $k=9$.