The binomial expansion of (0.2+0.8)^20 provides terms for all possibilities from all 20 women crying to no women crying. The expansion of the binomial expression has the coefficients 1, 20, 190, 1140, ... 20Cr, ... 190, 20, 1. The corresponding probability factors are 0.2^20, 0.2^19*0.8, 0.2^18*0.8^2, ... 0.2^(20-r)0.8^r, ... 0.2^2*0.8^18, 0.2*0.8^19, 0.8^20. These factors are: all 20 with crying response, 19 crying and 1 not crying, 18 crying and 2 not crying, and so on up to only 2 crying, only 1 crying, none crying.
a. At least 5 crying
This includes all women crying, i.e., we need to sum all the binomial terms from r=0 to 15. This comes to 0.3704.
b. 1 to 3
Sum from r=17 to 19 = 0.4000.
c. At most 6
Sum from 14 to 20=0.9133.