Question

At a certain conference of $100$ people there are $29$ Indian women and $23$ Indian men. Of these Indian people $4$ are doctors and $24$ are either men or doctors. There are no foreign doctors. How many foreigners are attending the conference? How many women doctors are attending the conference?

Answer 1

Let $E$ be the set of Indian people and $F$ the set of foreigners. $W$ stands for Women, $M$ for Men and $D$ for Doctors.
$E=W\cup M$
$\therefore n\left(E\right)=n(W\cup M)$

$=n\left(W\right)+n\left(M\right)-n(W\cap M)$
$=29+23-n\left(\varphi \right)$ $(W\cap M=\varphi$ as a man cannot be a woman.)

$=52-0=52$

Foreign delegates= Total-Indian $=100-52=48$
Again we are given $n(M\cup D)=24$
or $n\left(M\right)+n\left(D\right)-n(M\cap D)=24$
$23+4-n(M\cap D)=24$

$\therefore n(M\cap D)=3$

Hence amongst the set of men there are three doctors. Since the total number of doctors (Indian only) is $4$ out of which we have shown $3$ are men and hence we conclude there is only one woman (Indian) doctor. There are no foreign doctors.