Rahul, Rakshita and Gurmeet, working together, would have taken more than $7$ days to finish a job. On the other hand, Rahul and Gurmeet, working together would have taken less than $15$ days to finish the job. However, they all worked together for $6$ days, followed by Rakshita, who worked alone for $3$ more days to finish the job. If Rakshita had worked alone on the job then the number of days she would have taken to finish the job, cannot be

We need to set up equations based on the given conditions and find the constraints on Rakshita's working time.

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Let us define the variables clearly:

Rahul takes a days to complete the job alone

Rakshita takes b days to complete the job alone

Gurmeet takes c days to complete the job alone

Their work rates (portion of job completed per day):

Rahul's rate: $\frac{1}{a}$

Rakshita's rate: $\frac{1}{b}$

Gurmeet's rate: $\frac{1}{c}$

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Condition 1: "All three working together would take more than 7 days"

Combined rate of all three: $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

Time taken together: $\frac{1}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} > 7$

Therefore: $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} < \frac{1}{7}$ ... (1)

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Condition 2: "Rahul and Gurmeet together would take less than 15 days"

Combined rate of Rahul and Gurmeet: $\frac{1}{a} + \frac{1}{c}$

Time taken together: $\frac{1}{\frac{1}{a} + \frac{1}{c}} < 15$

Therefore: $\frac{1}{a} + \frac{1}{c} > \frac{1}{15}$ ... (2)

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Condition 3: "All worked together for 6 days, then Rakshita alone for 3 days to finish"

Work done in 6 days by all three: $6\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right)$

Work done in 3 days by Rakshita alone: $3\left(\frac{1}{b}\right)$

Total work = 1 complete job

Therefore: $6\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) + 3\left(\frac{1}{b}\right) = 1$ ... (3)

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From equation (3): $6\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) = 1 - \frac{3}{b}$

Using the upper bound from condition (1):

Since $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} < \frac{1}{7}$, we get:

$6\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) < \frac{6}{7}$

Therefore: $1 - \frac{3}{b} < \frac{6}{7}$

This gives us: $\frac{1}{7} < \frac{3}{b}$

Therefore: $b < 21$

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Using the lower bound from condition (2):

From equation (3): $6\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) + \frac{3}{b} = 1$

This can be rewritten as: $6\left(\frac{1}{a} + \frac{1}{c}\right) + \frac{6}{b} + \frac{3}{b} = 1$

Simplifying: $6\left(\frac{1}{a} + \frac{1}{c}\right) + \frac{9}{b} = 1$

Since $\frac{1}{a} + \frac{1}{c} > \frac{1}{15}$ from condition (2):

$6\left(\frac{1}{a} + \frac{1}{c}\right) > \frac{6}{15} = \frac{2}{5}$

So: $\frac{2}{5} + \frac{9}{b} < 1$

This gives us: $\frac{9}{b} < \frac{3}{5}$

Therefore: $b > 15$

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From our analysis:

$b < 21$ (from the upper constraint)

$b > 15$ (from the lower constraint)

So Rakshita would take between 15 and 21 days to complete the job alone.

Considering the typical answer choices for such problems, b cannot be 21 since our constraint shows $b < 21$.

The answer is 21 days.