Rewritten Solution

1. Key Concepts and Formulas

This problem requires the application of concepts from the Binomial Theorem and Coordinate Geometry.

• Binomial Theorem: The expansion of $(x+y)^N$ is given by $\sum_{k=0}^N \binom{N}{k} x^{N-k} y^k$.
• Sum of Binomial Coefficients: For an expansion of the form $(x+y)^N$, the sum of all coefficients is $2^N$. This can be obtained by substituting $x=1$ and $y=1$ into the expansion.
• Number of Terms in Binomial Expansion: The expansion of $(x+y)^N$ has $N+1$ terms (from $k=0$ to $k=N$).
• Arithmetic Mean: The arithmetic mean of a set of values is the sum of the values divided by the number of values.
• Distance of a Point from a Line: The perpendicular distance $d$ of a point $\mathrm{P}(x_1, y_1)$ from a line $Ax+By+C=0$ is given by the formula: $d = \frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$

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2. Step-by-step Working with Explanations

Step 1: Identify the degree of the binomial expansion and its coefficients. The given binomial expansion is $(x+y)^{2n-3}$. Here, the degree of the expansion, $N$, is $2n-3$. The coefficients in this expansion are $\binom{2n-3}{0}, \binom{2n-3}{1}, \dots, \binom{2n-3}{2n-3}$.

Step 2: Calculate the sum of all coefficients. Using the property that the sum of coefficients in the expansion of $(x+y)^N$ is $2^N$: Sum of coefficients, $S = \sum_{k=0}^{2n-3} \binom{2n-3}{k} = 2^{2n-3}$.

• Explanation: By setting $x=1$ and $y=1$ in the binomial expansion of $(x+y)^{2n-3}$, we obtain $(1+1)^{2n-3} = 2^{2n-3}$, which directly represents the sum of all coefficients.

Step 3: Determine the number of terms (coefficients) in the expansion. The number of terms in the expansion of $(x+y)^N$ is $N+1$. For $(x+y)^{2n-3}$, the number of terms is $(2n-3)+1 = 2n-2$.

• Explanation: Since the coefficients are indexed from $k=0$ to $k=2n-3$, there are $(2n-3) - 0 + 1 = 2n-2$ distinct coefficients, which is equal to the number of terms.

Step 4: Formulate the equation for the arithmetic mean. We are given that the arithmetic mean of all coefficients is 16. Using the definition of arithmetic mean: $\text{Mean} = \frac{\text{Sum of coefficients}}{\text{Number of coefficients}}$ $16 = \frac{2^{2n-3}}{2n-2}$

• Explanation: This step combines the results from Step 2 (sum of coefficients) and Step 3 (number of coefficients) with the given value of the mean to form an algebraic equation in terms of $n$.

Step 5: Solve the equation for the integer $n$. $16 = \frac{2^{2n-3}}{2n-2}$ Multiply both sides by $(2n-2)$: $16(2n-2) = 2^{2n-3}$ Express 16 as a power of 2: $16 = 2^4$. Also, factor out 2 from $(2n-2)$: $(2n-2) = 2(n-1)$. Substitute these into the equation: $2^4 \cdot 2(n-1) = 2^{2n-3}$ $2^5 (n-1) = 2^{2n-3}$ To find the integer solution for $n \geq 2$, we can test values:

• If $n=2$: $2^5 (2-1) = 32 \cdot 1 = 32$. $2^{2(2)-3} = 2^1 = 2$. ($32 \neq 2$)
• If $n=3$: $2^5 (3-1) = 32 \cdot 2 = 64$. $2^{2(3)-3} = 2^3 = 8$. ($64 \neq 8$)
• If $n=4$: $2^5 (4-1) = 32 \cdot 3 = 96$. $2^{2(4)-3} = 2^5 = 32$. ($96 \neq 32$)
• If $n=5$: $2^5 (5-1) = 32 \cdot 4 = 128$. $2^{2(5)-3} = 2^{10-3} = 2^7 = 128$. ($128 = 128$) Thus, the integer value for $n$ is 5.
• Explanation: We simplify the equation using properties of exponents. Since this is a mixed equation (exponential and polynomial), analytical solution might be complex. Testing integer values, constrained by $n \geq 2$, is an effective method to find the correct $n$.

Step 6: Determine the coordinates of point P. The point P is given by $(2n-1, n^2-4n)$. Substitute $n=5$ into the coordinates: $x$-coordinate: $2(5)-1 = 10-1 = 9$ $y$-coordinate: $5^2 - 4(5) = 25 - 20 = 5$ So, the coordinates of point P are $(9, 5)$.

• Explanation: We simply substitute the value of $n$ found in the previous step into the algebraic expressions defining the coordinates of point P.

Step 7: Calculate the distance of point P from the line $x+y=8$. The point is $\mathrm{P}(9, 5)$. The equation of the line is $x+y=8$, which can be rewritten in the standard form $Ax+By+C=0$ as $x+y-8=0$. Here, $A=1$, $B=1$, and $C=-8$. Using the distance formula $d = \frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$: $d = \frac{|(1)(9) + (1)(5) - 8|}{\sqrt{1^2 + 1^2}}$ $d = \frac{|9 + 5 - 8|}{\sqrt{1 + 1}}$ $d = \frac{|14 - 8|}{\sqrt{2}}$ $d = \frac{|6|}{\sqrt{2}}$ $d = \frac{6}{\sqrt{2}}$ To rationalize the denominator, multiply the numerator and denominator by $\sqrt{2}$: $d = \frac{6 \sqrt{2}}{2}$ $d = 3\sqrt{2}$

• Explanation: We apply the standard formula for the perpendicular distance from a point to a line. The absolute value ensures the distance is non-negative. Rationalizing the denominator provides a simplified and conventional form of the answer.

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3. Tips and Common Mistakes

• Careful with Binomial Degree: Ensure you correctly identify $N$ (the power) in the binomial expansion to apply formulas for sum of coefficients and number of terms correctly.
• Solving Mixed Equations: Equations involving both powers and linear terms (like $2^k = C \cdot k$) often require trial and error for integer solutions, or graphical methods for general solutions.
• Distance Formula: Remember to rewrite the line equation in the $Ax+By+C=0$ form before identifying $A, B, C$. Don't forget the absolute value in the numerator and the square root in the denominator.
• Rationalizing Denominators: Always simplify expressions by rationalizing denominators when they involve square roots.

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4. Summary and Key Takeaway

This problem effectively tests understanding of fundamental properties of binomial expansions, particularly the sum and number of coefficients, combined with the geometric concept of the distance from a point to a line. The crucial steps involved setting up an equation for $n$ using the given arithmetic mean, solving it, determining the point's coordinates, and finally applying the distance formula. The final calculated distance is $3\sqrt{2}$.

(Note: Based on standard mathematical formulas and the problem statement, the derived answer is $3\sqrt{2}$, which corresponds to option (D). If option (A) $\sqrt{2}$ is truly the correct answer, there might be a discrepancy in the problem statement's given mean value or the options provided.)