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Review

Review. Previously we determined the number of r-permutations of a set, P(n, r), and we found a formula to calculate P(n, r): And, we determined the number of r-combinations of a set, C(n, r), and we found a formula to calculate C(n, r): And all of this lead to the Binomial Theorem:.

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Review

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  1. Review • Previously we determined the number of r-permutations of a set, P(n, r), and we found a formula to calculate P(n, r): • And, we determined the number of r-combinations of a set, C(n, r), and we found a formula to calculate C(n, r): • And all of this lead to the Binomial Theorem:

  2. Permutations with Repetition • How many strings of length 8 can be formed with the English alphabet? • Product Rule: 26 letters….268 • Theorem 1: The number of r-permutations of a set of n objects with repetition allowed is nr

  3. Combinations with Repetition • Assume there are at least four of each of the following in a bowl: • Apples • Oranges • Pears • How many ways are there to select four pieces of fruit? • Solution: • n = 3 objects in the set, and we are looking for the number of r-permutations with r=4: • Theorem 2: There are C(n+r-1,r) r-combinations from a set with n elements when repetition of elements is allowed

  4. Combinations with Repetition • Assume you have the following denominations (bills) in a cash box: • $100, $50, $20, $10, $5, $2, $1 • How many ways to select five bills? Assume order doesn’t matter and that there are at least five of each type. • C(n+r-1,r) r-combinations from a set with n elements when repetition of elements is allowed

  5. Combinations with Repetition • Assume you have four different types of cookies in a shop. How many ways are there to choose six? • C(n+r-1,r) r-combinations from a set with n elements when repetition of elements is allowed

  6. Combinations with Repetition • Assume you have four different types of cookies in a shop. How many ways are there to choose six? • C(n+r-1,r) r-combinations from a set with n elements when repetition of elements is allowed

  7. Summary • r-Permutations, repetition not allowed: • r-Combinations, repetition not allowed: • r-Permutations, repetition allowed: • r-Combinations, repetition allowed:

  8. Review for Exam • Product Rule • Sum Rule • Pigeonhole Principle • Generalized Pigeonhole Principle • Permutations • Combinations • Binomial Coefficients • Pascal’s Triangle • Generalized Permutations and Combinations

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