Assume the total population of Panchagarh state in 2016 is 350000. Then find the average female population of cities P, Q, and S in 2016.

The total population is given as 350000. 

It is also given that 20% of the total population resides in city P.

So, the number of people in city P = $$\ \dfrac{\ 20}{100}\times\ 350000\ =70000\ $$

Now, it is given that 20% of the population is male. So the female population would be equal to $$\ \ \left(1-\ \dfrac{\ 20}{100}\right)\%\ =80\%$$
Number of females = $$\ \ \ \dfrac{\ 80}{100}\times\ 70000\ =56000$$

Similarly, It is given that 28% of the total population resides in city Q.

So, the number of people in city Q = $$\ \dfrac{\ 28}{100}\times\ 350000\ =98000\ $$

It is given that 65% of the population is male. So the female population would be equal to $$\ \ \left(1-\ \dfrac{\ 65}{100}\right)\%\ =35\%$$

Number of females = $$\ \ \ \dfrac{\ 35}{100}\times\ 98000\ =34300$$

Moreover, 

It is given that 15% of the total population resides in city S.

So, the number of people in city S $$\ \dfrac{\ 15}{100}\times\ 350000\ =52500\ $$

It is given that 50% of the population is male. So the female population would be equal to $$\ \ \left(1-\ \frac{\ 50}{100}\right)\%\ =50\%$$

Number of females in S = $$\ \ \ \dfrac{\ 50}{100}\times\ 52500\ =26250$$

Average female population of cities P, Q, and S is equal to $$\ \ \ \ \ \dfrac{\ 56000+34300+26250}{3}\ =38,850$$