A certain club is forming a recruitment committee consisting of five of its members. They have calculated that there are 8,568 different ways to form such a committee. The club has two members named Jack and Jill. They have calculated that 2,380 of the potential committees have Jack on them, 2,380 have Jill, 1,820 have Jack but not Jill, 1,820 have Jill but not Jack, and 560 have both Jack and Jill. Let U be the set of all possible committees of five that can be formed; let A be the set of these committees that have Jack on it, let B be the set of these committees that have Jill on it. Represent the above given information using set cardinality, for example: |U |=8,568 How many committees have either Jack or Jill on it? How many committees have neither Jack nor Jill on it? Jack and Jill are car-pooling, so they insist that IF either one is on the committee, the other person must also be on the committee. How many committees meet this condition? Jack and Jill have had a fight. Jack says, "If Jill is on the committee, I won't be." Jill says, "Likewise." How many committees meet this condition?
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.
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A certain club is forming a recruitment committee consisting of five of its members. They have calculated that there are 8,568 different ways to form such a committee. The club has two members named Jack and Jill. They have calculated that 2,380 of the potential committees have Jack on them,
2,380 have Jill, 1,820 have Jack but not Jill, 1,820 have Jill but not Jack, and 560 have both Jack and Jill.
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Let U be the set of all possible committees of five that can be formed; let A be the set of these committees that have Jack on it, let B be the set of these committees that have Jill on it. Represent the above given information using set cardinality, for example:
|U |=8,568
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How many committees have either Jack or Jill on it?
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How many committees have neither Jack nor Jill on it?
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Jack and Jill are car-pooling, so they insist that IF either one is on the committee, the other person must also be on the committee. How many committees meet this condition?
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Jack and Jill have had a fight. Jack says, "If Jill is on the committee, I won't be." Jill says, "Likewise." How many committees meet this condition?
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