Correct Answer - Option 3 : Two

**CONCEPT**:

- The direction ratios of the line joining the points (x
_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) is given by: a = x_{2} - x_{1}, b = y_{2} - y_{1} and c = z_{2} - z_{1}
- If a, b, c are the direction ration ratios of a line passing through the point (x
_{1}, y_{1}, z_{1}), then the equation of line is given by: \(\frac{{x - {x_1}}}{a} = \frac{{y - {y_1}}}{b} = \frac{{z - {z_1}}}{c}\)

**CALCULATION**:

Given: The line joining the points (k, 1, 3) and (1, -2, k + 1) also passes through the point (15, 2, -4)

As we know that, the direction ratios of the line joining the points (x1, y1, z1) and (x2, y2, z2) is given by: a = x2 - x1, b = y2 - y1 and c = z2 - z_{1}

So, the direction ratios of the line joining the points (k, 1, 3) and (1, -2, k + 1) is: a = 1 - k, b = - 3 and c = k - 2

As we know that, if a, b, c are the direction ration ratios of a line passing through the point (x1, y1, z1), then the equation of line is given by: \(\frac{{x - {x_1}}}{a} = \frac{{y - {y_1}}}{b} = \frac{{z - {z_1}}}{c}\)

So, the equation of the line with direction ratios a, b, c and passing through the the point (k, 1, 3) is given by: \(\frac{{x - k}}{1 - k} = \frac{{y - 1}}{- 3} = \frac{{z - 3}}{k - 2}\)

∵ It is given that, the line represented by \(\frac{{x - k}}{1 - k} = \frac{{y - 1}}{- 3} = \frac{{z - 3}}{k - 2}\) also passes through the point (15, 2, -4)

So, substitute x = 15, y = 2 and z = - 4 in the equation \(\frac{{x - k}}{1 - k} = \frac{{y - 1}}{- 3} = \frac{{z - 3}}{k - 2}\)

⇒ \(\frac{{15 - k}}{1 - k} = \frac{{2 - 1}}{- 3} = \frac{{-4 - 3}}{k - 2}\)

⇒ \(\frac{{15 - k}}{1 - k} = \frac{{1}}{- 3} \)

⇒ - 45 + 3k = 1 - k

⇒ k = 23/2

⇒ \(\frac{{1}}{- 3} = \frac{{-7}}{k - 2}\)

⇒ k - 2 = 21

⇒ k = 23

So, there are two possible values of k as shown above.

Hence, correct option is 3