The executive of an organization is to be selected from a group of 6 males and 6

Question

The executive of an organization is to be selected from a group of 6 males and 6 females.

a) How many ways are there to choose the president and secretary?

b) How many ways are there to have a female president and a male secretary?
c) What is the probability that a randomly chosen president is female and a randomly chosen secretary is male?

d) How may ways are there to choose a president, secretary and treasurer?

e) How many ways are there to choose a president, secretary and treasurer so that two are male and one is female?

f) What is the probability that a randomly chosen president, secretary and treasurer include two males and one female?

Explanation / Answer

a. ways are there to choose the president and secretary = 12C2 (Combination of choosing two people from 12 present = 12! / (2! * 10!) = 66. This should be multiplied by 2 since one can be a president in one combination and again secetary in another combination = 132 is the answer

b. ways are there to have a female president and a male secretary(both independent events) = ways of choosing a president out of 6 females * ways of choosing a secretary out of 6 males = 6C1 * 6C1 = 2* ( 6!/ (5!*1!)) = 12

c. probability that a randomly chosen president is female and a randomly chosen secretary is male = Probability of choosing a female * probability of choosing a male without replacing the previously selected female president = 6/12 * 6/11 = 0.273

d. ways are there to choose a president, secretary and treasurer = Combination scenario of choosing 3 out of 12 people = 12C3 = 12! / (9!*3!) = 220

e. ways are there to choose a president, secretary and treasurer = An extension of option C where we don't replace any candidates once selected as a candidate can't hold more than one position = ways such that we have a male president, male secretary and female treasurer + ways such that we have a female president, male secretary and male treasurer + ways such that we have a male president, female secretary and male treasurer = 6C2 * 6C1 * 6 (Three posts can have 6 ways of arranging between themselves) = 540

f. the probability that a randomly chosen president, secretary and treasurer include two males and one female = ways are there to choose a president, secretary and treasurer so that two are male and one is female / total number of ways three posts can be filled up by anyone = 540 / (12C3* 6) = 0.2045

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