Exercise 1.A committee of five people is chosen randomly from four men and six women. Find the probability that:a) Exactly four women are on the committee:b) At least four women are on the committee:c) At most four women are on the committee:

Answer with explanation:Given : Number of men = 4Number of women = 6Total people = [tex]6+4=10[/tex]Total number of ways to make a committee of five people from 10 persons :-[tex]^{10}C_{4}=\dfrac{10!}{4!(10-4)!}=210[/tex]a) Number of ways to make committee that has exactly four women :[tex]^6C_4\times ^4C_1=\dfrac{6!}{4!(6-4)!}\times4=60[/tex]The  probability that committee has exactly four women :[tex]\dfrac{60}{210}=\dfrac{2}{7}[/tex]b) Number of ways to make committee that has at-least four women :[tex]^6C_4\times ^4C_1+^6C_5=\dfrac{6!}{4!(6-4)!}\times4+6=66[/tex]The probability that committee at-least four women :[tex]\dfrac{66}{210}=\dfrac{22}{70}[/tex]c) Number of ways that committee has more than 4 women :-[tex]^6C_5\times^4C_0=6[/tex]The probability that committee has more than 4 women :-[tex]\dfrac{6}{210}[/tex]Now, the  probability that committee has at most four women :-[tex]1-\dfrac{6}{210}=\dfrac{204}{210}=\dfrac{102}{105}[/tex]