Correct Option

To determine the value of 'n', we first calculate the individual work rates of X, Y, and Z, and then use these rates to establish the total work equation.

X completes 1/3 of the work in 6 days. Therefore, X can complete the entire work in 6 × 3 = 18 days. X's one-day work rate is 1/18.
Y completes 1/3 of the work in 8 days. Therefore, Y can complete the entire work in 8 × 3 = 24 days. Y's one-day work rate is 1/24.
Z completes 3/4 of the work in 12 days. Therefore, Z can complete the entire work in 12 × (4/3) = 16 days. Z's one-day work rate is 1/16.

The combined one-day work rate of X, Y, and Z is the sum of their individual rates:

Combined rate (X+Y+Z) = (1/18) + (1/24) + (1/16)

To add these fractions, the Least Common Multiple (LCM) of 18, 24, and 16 is 144.

Combined rate (X+Y+Z) = (8/144) + (6/144) + (9/144) = (8 + 6 + 9)/144 = 23/144 units of work per day.

According to the problem statement:

X, Y, and Z work together for 'n' days. The work done in this period is n × (23/144).
Subsequently, X and Z quit, and Y alone finishes the remaining work in 13/3 days.
The work done by Y alone during this period is (13/3) × (1/24) = 13/72 units of work.

Since the total work is 1 unit, we can set up the equation:

n × (23/144) + 13/72 = 1

Now, solve for 'n':

23n/144 = 1 - 13/72

23n/144 = (72 - 13)/72

23n/144 = 59/72

Multiply both sides by 144:

23n = (59/72) × 144

23n = 59 × 2

23n = 118

n = 118/23

n = 4

Therefore, X, Y, and Z worked together for 4 days.

Incorrect Options:

Options (1) 3, (3) 5, and (4) 6 are incorrect as they do not satisfy the derived value of n = 4 days based on the calculated work rates and the total work completed.