conditional probability : planning and development questions
Your probability of $P(E \cap F) = \dfrac 17$ is correct.
If $e$ students opt for English language workshop, $f$ students opt for French and $g$ students for both,
$e + f - g = 21 \tag1$
Now using conditional probability we know that,
$ \displaystyle \frac{g}{e} = \frac 15 \implies \displaystyle e = 5g$
Similarly, $ \displaystyle \frac{g}{f} = \frac 13 \implies f = 3g$
Plugging into $(1)$, $ \displaystyle 5g + 3g - g = 21$
$ \implies g = 3. ~$ So there are $3$ students who have opted for both English and French language workshops.