The question asks us to find the coefficient of a specific term, $x^{101}y^{99}$, in the binomial expansion of $(2x - 3y)^{200}$. To solve this, we will use the Binomial Theorem.

Understanding Binomial Expansion and the General Term

The Binomial Theorem provides a formula for expanding expressions of the form $(a+b)^n$. The general term in the expansion of $(a+b)^n$ is given by:

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

Here, $n$ is the power to which the binomial is raised, $r$ is an index starting from 0, $a$ is the first term, and $b$ is the second term of the binomial.

Applying the Binomial Theorem to (2x - 3y)²⁰⁰

In our case, the expression is $(2x - 3y)^{200}$. Comparing this with the general form $(a+b)^n$, we have:

• $n = 200$
• $a = 2x$
• $b = -3y$

Substituting these values into the general term formula, we get the general term for the expansion of $(2x - 3y)^{200}$:

$$ T_{r+1} = \binom{200}{r} (2x)^{200-r} (-3y)^r $$

Now, let's expand the terms $(2x)^{200-r}$ and $(-3y)^r$:

$$ (2x)^{200-r} = 2^{200-r} x^{200-r} $$

$$ (-3y)^r = (-3)^r y^r $$

Substitute these back into the general term formula:

$$ T_{r+1} = \binom{200}{r} (2^{200-r} x^{200-r}) ((-3)^r y^r) $$

Rearranging the terms to group coefficients and variables:

$$ T_{r+1} = \binom{200}{r} 2^{200-r} (-3)^r x^{200-r} y^r $$

Finding the Value of r for the Desired Term

We are looking for the term with $x^{101}y^{99}$. Comparing the powers of $x$ and $y$ in the general term ($x^{200-r} y^r$) with the powers in the desired term ($x^{101}y^{99}$), we can find the value of $r$.

• Power of x: $200 - r = 101$
• Power of y: $r = 99$

From the power of y, we directly get $r = 99$. Let's check if this value of $r$ is consistent with the power of x:

$$ 200 - r = 101 $$

$$ 200 - 99 = 101 $$

$$ 101 = 101 $$

Yes, the value $r=99$ is consistent for both x and y powers. This means the term with $x^{101}y^{99}$ is the term when $r=99$, which is $T_{99+1} = T_{100}$.

Calculating the Coefficient of x¹⁰¹y⁹⁹

The coefficient of the term $x^{200-r} y^r$ in the general term is given by $\binom{200}{r} 2^{200-r} (-3)^r$. We need to find this coefficient when $r=99$.

Substitute $r=99$ into the coefficient part:

Coefficient = $\binom{200}{99} 2^{200-99} (-3)^{99}$

Simplify the exponents:

Coefficient = $\binom{200}{99} 2^{101} (-3)^{99}$

Since 99 is an odd number, $(-3)^{99}$ can be written as $(-1 \cdot 3)^{99} = (-1)^{99} \cdot 3^{99} = -1 \cdot 3^{99} = -3^{99}$.

Substitute this back into the coefficient:

Coefficient = $\binom{200}{99} 2^{101} (-3^{99})$

Coefficient = $-\binom{200}{99} 2^{101} 3^{99}$

This is the required coefficient of $x^{101}y^{99}$ in the expansion of $(2x - 3y)^{200}$.

Let's compare this with the given options.

The calculated coefficient is \(-\left( {\begin{array}{*{20}{c}} {200}\\ {99} \end{array}} \right){2^{101}}{3^{99}}\).

The calculated coefficient matches Option 1.

Revision Table: Key Concepts

Additional Information: Properties of Coefficients in Binomial Expansion

When dealing with binomial expansion problems, several properties can be useful:

• Sum of Coefficients: The sum of the coefficients in the expansion of $(a+b)^n$ is found by setting $a=1$ and $b=1$. So, the sum is $(1+1)^n = 2^n$.
• Alternating Signs: If the binomial is of the form $(a-b)^n$, the signs of the terms will alternate. This is because the second term $b$ is negative. The general term involves $(-b)^r$, which results in a positive sign if $r$ is even and a negative sign if $r$ is odd.
• Symmetry of Binomial Coefficients: $\binom{n}{r} = \binom{n}{n-r}$. This means coefficients equidistant from the beginning and end of the expansion are equal.
• Middle Term: The middle term(s) in a binomial expansion $(a+b)^n$ depend on whether $n$ is even or odd.
  • If $n$ is even, there is one middle term: $T_{n/2 + 1}$ (where $r = n/2$).
  • If $n$ is odd, there are two middle terms: $T_{(n+1)/2}$ (where $r = (n-1)/2$) and $T_{(n+3)/2}$ (where $r = (n+1)/2$).

In our problem, the term was $(2x - 3y)^{200}$, where the second term is negative. The power of the second term $(-3y)$ in the term $x^{101}y^{99}$ was 99, which is odd. This correctly led to a negative sign in the coefficient.