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

The correct option is 21.

Let the rate of work done per day by Rahul, Rakshita, and Gurmeet be $r$, $k$, and $g$ respectively. Let the total work to be completed be $1$ unit.

Step 1: Set up equations and inequalities from the given statements
1. All three working together would have taken more than 7 days to finish the job:
$\frac{1}{r+k+g}>7\Rightarrow r+k+g<\frac{1}{7}$      --- (Inequality 1)

2. Rahul and Gurmeet working together would have taken less than 15 days to finish the job:
$\frac{1}{r+g}<15\Rightarrow r+g>\frac{1}{15}$      --- (Inequality 2)

3. They all worked together for 6 days, and then Rakshita worked alone for 3 more days to complete the job:
$6(r+k+g)+3k=1$
Simplifying this equation:
$6(r+g)+9k=1$
We can express $r+g$ in terms of $k$:
$r+g=\frac{1-9k}{6}$      --- (Equation 3)

Step 2: Determine the bounds for Rakshita's rate of work ($k$)
Using Inequality 1:
$(r+g)+k<\frac{1}{7}$
Substituting Equation 3 into Inequality 1:
$\frac{1-9k}{6}+k<\frac{1}{7}$
Multiply the inequality by 6:
$1-9k+6k<\frac{6}{7}$
$1-3k<\frac{6}{7}$
$3k>1-\frac{6}{7}$
$3k>\frac{1}{7}\Rightarrow k>\frac{1}{21}$      --- (Bound 1)

Next, using Inequality 2:
$r+g>\frac{1}{15}$
Substituting Equation 3 into Inequality 2:
$\frac{1-9k}{6}>\frac{1}{15}$
Multiply the inequality by 6:
$1-9k>\frac{6}{15}$
$1-9k>\frac{2}{5}$
$9k<1-\frac{2}{5}$
$9k<\frac{3}{5}\Rightarrow k<\frac{3}{45}\Rightarrow k<\frac{1}{15}$      --- (Bound 2)

Combining both bounds, we get:
$\frac{1}{21}<k<\frac{1}{15}$

Step 3: Find the bounds for the number of days taken by Rakshita alone
If Rakshita works alone, the number of days she would take to complete the job is $D=\frac{1}{k}$.
Taking the reciprocal of the inequality:
$15<D<21$
Thus, the number of days Rakshita takes to finish the job alone must be strictly between 15 and 21.

Since $D$ must be strictly less than 21, the number of days she would take cannot be 21.