In how many ways can a committee of three be chosen from 10 female and 20 male? if
a) All are equally eligible?
b) One female?
c) All are female?
d) Two female?
e) What is probability of selecting one female in a committee?
f) What is probability of selecting two female in a committee?
g) What is probability of selecting three female in a committee? �
a) To choose a committee of three from a total of 10 females and 20 males, we can use the combination formula. The number of ways to choose three people from a total of 30 is given by:
C(30, 3) = 30! / (3!(30 - 3)!) = 30! / (3!27!) = (30 * 29 * 28) / (3 * 2 * 1) = 4060
Therefore, there are 4060 ways to choose a committee of three from the given group.
b) To choose a committee of three with one female, we need to choose one female from the 10 available females and two males from the 20 available males. The number of ways to do this is given by:
C(10, 1) * C(20, 2) = (10 * 9 * 8) / (1 * 2 * 3) * (20 * 19) / (2 * 1) = 10 * 3 * 20 * 19 = 11,400
Therefore, there are 11,400 ways to choose a committee of three with one female.
c) To choose a committee of three with all females, we need to choose three females from the 10 available females. The number of ways to do this is given by:
C(10, 3) = (10 * 9 * 8) / (3 * 2 * 1) = 120
Therefore, there are 120 ways to choose a committee of three with all females.
d) To choose a committee of three with two females, we need to choose two females from the 10 available females and one male from the 20 available males. The number of ways to do this is given by:
C(10, 2) * C(20, 1) = (10 * 9) / (2 * 1) * 20 = 450
Therefore, there are 450 ways to choose a committee of three with two females.
e) The probability of selecting one female in a committee is given by the number of ways to select one female divided by the total number of possible committees. From part b), the number of ways to select one female is 11,400. From part a), the total number of possible committees is 4060. Therefore, the probability of selecting one female is:
11,400 / 4060 = 2.8104
f) The probability of selecting two females in a committee is given by the number of ways to select two females divided by the total number of possible committees. From part d), the number of ways to select two females is 450. From part a), the total number of possible committees is 4060. Therefore, the probability of selecting two females is:
450 / 4060 = 0.1108
g) The probability of selecting three females in a committee is given by the number of ways to select three females divided by the total number of possible committees. From part c), the number of ways to select three females is 120. From part a), the total number of possible committees is 4060. Therefore, the probability of selecting three females is:
120 / 4060 = 0.0296