A) A jury of 9 members is to be selected from a group of 25 individuals. How many different possible juries can be formed from this group? If the 25 individuals include 15 men and 10 women, how many of the possible juries would have 5 men and 4 women? 6 men and 3 women? Suppose th 25 individuals can be partitioned as 10 democrats, 9 republicans, and 6 independents. How many of the juries would have 3 democrats, 3 republicans, and 3 independents?
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
A) A jury of 9 members is to be selected from a group of 25 individuals.
How many different possible juries can be formed from this group?
If the 25 individuals include 15 men and 10 women, how many of the possible juries would have 5 men and 4 women? 6 men and 3 women?
Suppose th 25 individuals can be partitioned as 10 democrats, 9 republicans, and 6 independents.
How many of the juries would have 3 democrats, 3 republicans, and 3 independents?
B) There are 8 horses (A, B, C, D, E, F, G, and H) in a horse race.
The results of the horse race are given with first, second, and third place.
If I choose 3 horses at random to choose as first, second, and third place, what is the probability that I guess the results of the race correctly?
c)
Suppose there are 30 candidates who apply for a position and 4 are to be selected to be invited for an onsite interview.
How many different groups of 4 can be selected?
Suppose 18 of the applicants are men and 12 are women.
Also, suppose the 4applicants are selected at random.
What is the probability the 4 applicants selected include 2 men and 2 women? 3 men and 1 woman?
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