Question

The line \(y = 2x+c\) passes through a point that is equidistant from both the axes and lies in the first quadrant \((x > 0,y > 0)\). Then the value of c is...

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(-2\)

Hint:

The 7 identical balls count as one kind; select any number of them and fill the rest from 16 distinct balls.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
There are 7 identical balls and 16 distinct balls. If \(r\) identical balls are taken (\(0 \le r \le 7\)), there is only 1 way to choose them, and the other \(12 - r\) balls come from the 16 distinct ones.

Step 2: Add up the cases:
\[ N = \sum_{r=0}^{7} {}^{16}C_{12-r} = {}^{16}C_{12} + {}^{16}C_{11} + \dots + {}^{16}C_{5} \]
The values are 1820, 4368, 8008, 11440, 12870, 11440, 8008 and 4368, and their sum is 62322.

Step 3: Match with the options:
Option (A): \({}^{18}C_6 + {}^{18}C_8 = 18564 + 43758 = 62322\). This equals the count. Other options give \(75582 + 31824 = 107406\), \(12870 + 816 = 13686\) and \(12870 + 48620 = 61490\), which do not match.

Final Answer:
The number of selections is \({}^{18}C_6 + {}^{18}C_8 = 62322\), option (A). \[ \boxed{{}^{18}C_6 + {}^{18}C_8} \]