Question

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is: \\ \\

• A. \(580\)
• B. \(578\)
• C. \(576\)
• D. \(582\)

Hint:

\begin{tcolorbox}[colback=white, colframe=black, colbacktitle=white, coltitle=black, title=QuickTip, width=\textwidth, halign=left] The distance from a point to a line can be found by first finding the point of intersection and then using the distance formula. \end{tcolorbox} \vspace{0.5cm} \hline \vspace{0.5cm}

Answer 1

The Correct Option is A

% Solution\textbf{Solution:} \\\textbf{Step 1: Parameterize the line L.} \\The equation of line L passing through \( (0, 1, 2) \) is:\[\frac{x-0}{1} = \frac{y-1}{1} = \frac{z-2}{1} = \mu\]Thus, the coordinates of any point on line L are:\[(x, y, z) = (\mu, 1 + \mu, 2 + \mu)\]\textbf{Step 2: Find the point of intersection Q.} \\The equation of the given line is:\[\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}\]Let \( \mu = -3 \) to get the coordinates of the point of intersection \( Q' = (-3, -2, 1) \).\textbf{Step 3: Calculate the distance from point P to line L.} \\The distance between point \( P(1, -9, 2) \) and point \( Q'(-3, -2, 1) \) is:\[d = \sqrt{(16 + 49 + 9)} = \sqrt{74}\]\textbf{Final Answer:} The distance from P to L is \( \sqrt{74} \).