from a group of 6 men and 7 women going on a trip, find the number of
a) ways of selecting a female tour guide, a female cook and a driver of either sex
b) ways of selecting a commitee of 5 people with 4 or more men
7 ways for tour guide, then 6 ways for cook then 11 ways for driver
7 * 6 * 11
combinations of 5 out of the six men at a time
6!/[ 5!(1!)] = 6
combinations of 4 out of the 6 men
6!/[ 4!(2!)] = 6*5/2 = 15
If we have 4 men, then one of the 7 women must serve (we have already included the remaining two men in the five men option)
so for each combination of 4 men we have a choice of 7 women
so in all
6 + 15 * 7
since there is a title for a job so
(7 choose 2)(11 choose 1)
11 because u already choose 2 women
To solve these problems, we will use the concept of permutations and combinations.
a) To find the number of ways of selecting a female tour guide, a female cook, and a driver of either sex, we have to consider two cases:
Case 1: When the driver is male.
In this case, we have 6 men and 7 women to choose from for the tour guide and cook. We can select a female tour guide in 7 ways, a female cook in 6 ways, and the male driver in 6 ways. To calculate the total number of ways in this case, we multiply these values together: 7 x 6 x 6 = 252.
Case 2: When the driver is female.
In this case, we again have 6 men and 7 women to choose from for the tour guide and cook. We can select a female tour guide in 7 ways, a female cook in 6 ways, and a female driver in 7 ways. To calculate the total number of ways in this case, we multiply these values together: 7 x 6 x 7 = 294.
Finally, to find the total number of ways, we sum up the results from both cases: 252 + 294 = 546.
Therefore, there are 546 ways of selecting a female tour guide, a female cook, and a driver of either sex.
b) To find the number of ways of selecting a committee of 5 people with 4 or more men, we need to consider two cases:
Case 1: When there are exactly 4 men in the committee.
In this case, we can choose 4 men out of the 6 available in 6C4 ways (denoted as 6 choose 4). Additionally, we need to choose 1 woman out of the 7 available, which can be done in 7C1 ways. Therefore, the total number of ways in this case is 6C4 x 7C1 = 6 x 7 = 42.
Case 2: When all 5 people in the committee are men.
In this case, we can choose 5 men out of the 6 available in 6C5 ways. Therefore, the total number of ways in this case is 6C5 = 6.
Finally, to find the total number of ways, we sum up the results from both cases: 42 + 6 = 48.
Therefore, there are 48 ways of selecting a committee of 5 people with 4 or more men.