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Understanding Counting Techniques and Probability in Selections

This chapter explores fundamental counting techniques, including the Multiplication Principle of Counting. It explains how to calculate the number of different arrangements in sequential choices and permutations, such as license plate configurations and planting arrangements. The concepts of combinations and their application in selecting committee chairs or members are also discussed, alongside practical examples to illustrate these principles. Additionally, the chapter addresses the probability of drawing specific hands from a deck of cards, such as the chances of obtaining three of a kind.

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Understanding Counting Techniques and Probability in Selections

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  1. Chapter 5Probability 5.5 Counting Techniques

  2. Multiplication Principle of Counting If a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, and so on. Then the task of making these selections can be done in the following number of different ways.

  3. If a license plate consists of a letter, then 5 numbers, how many different types of license plates are possible?

  4. If a license plate consists of a letter, then 5 numbers, how many different types of license plates are possible?

  5. Suppose that you have 5 different trees that you wish to plant. In how many different ways can the 5 trees be planted in a row?

  6. A permutation is an ordered arrangement of n distinct objects without repetitions. The symbol nPr represents the number of permutations of n distinct objects, taken r at a time,where r<n.

  7. Determine the value of 9P3.

  8. Suppose a committee consists of 10 people. In how many ways can the chair and vice-chair of the committee be chosen if the first person randomly selected is the chair and the second person randomly selected is the vice-chair.

  9. A combination is an arrangement, without regard to order, of n distinct objects without repetitions. The symbol nCr represents the number of combinations of n distinct objects taken r at a time, where r<n.

  10. Determine the value of 9C3.

  11. The United States Senate consists of 100 members. In how many ways can 4 members be randomly selected to attend a luncheon at the White House?

  12. A landscape architect has 3 Sugar Maples, 2 Weeping Willows, and 4 Green Ash trees to plant along a street in a new subdivision. In how many different ways can the trees be planted?

  13. What is the probability of obtaining 3 of a kind when 5 cards are drawn from a standard 52-card deck?

  14. What is the probability of obtaining 3 of a kind when 5 cards are drawn from a standard 52-card deck? P(3 of a kind) = 0.0211

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