Combinatorial Principles, Permutations, and Combinations Section 03 Combinatorial Principles, Permutations, and Combinations

Permutations vs. Combinations Permutations are ordered Combinations are not ordered Therefore, there are more permutations than combinations for given 𝑛 and 𝑘 Both apply to combinatorics without replacement Also: remember 0! = 1

Permutations ORDER MATTERS Choosing an ordered subset of size 𝑘 from a collection of 𝑛 objects without replacement: 𝑛𝑃𝑘= 𝑛! 𝑛−𝑘 ! Given 𝑛 objects, of which 𝑛 1 are Type 1, 𝑛 2 are Type 2, etc, up to 𝑛 𝑡 , the number of ways to order all 𝑛 objects is: 𝑛! 𝑛 1 !∗ 𝑛 2 !∗…∗ 𝑛 𝑡 !

Combinations ORDER DOES NOT MATTER Choosing a subset of size 𝑘 from a collection of 𝑛 objects without replacement: 𝑛 𝑘 =𝑛𝐶𝑘= 𝑛! 𝑘!∗ 𝑛−𝑘 ! 𝑛 𝑘 is also called a binomial coefficient

Binomial Theorem 𝑛 𝑘 is called the binomial coefficient because it is used in the power series expansion of (1+𝑡) 𝑛 (1+𝑡) 𝑛 = 𝑘=0 ∞ 𝑁 𝑘 ∗ 𝑡 𝑘 = 1+𝑁𝑡+ 𝑁(𝑁−1) 2 𝑡 2 + 𝑁 𝑁−1 (𝑁−2) 6 𝑡 3 +… If N is as integer, summations stops at k=N If N is not an integer, series is only valid if -1<t<1 This expansion is useful for understanding the binomial distribution

Multinomial Theorem In the power series expansion of ( 𝑡 1 + 𝑡 2 +…+ 𝑡 𝑠 ) 𝑁 the coefficient of 𝑡 1 𝑛 1 ∗ 𝑡 2 𝑛 2 ∗…∗ 𝑡 𝑠 𝑛 is 𝑁 𝑛 1 𝑛 2 … 𝑛 𝑠 = 𝑁! 𝑛 1 !∗ 𝑛 2 !∗…∗ 𝑛 𝑠 ! Useful for understanding multinomial distributions (much later)

In conclusion… Ordered? Not ordered? 𝑛𝑃𝑘= 𝑛! 𝑛−𝑘 ! 𝑛𝑃𝑘= 𝑛! 𝑛−𝑘 ! Not ordered? 𝑛 𝑘 =𝑛𝐶𝑘= 𝑛! 𝑘!∗ 𝑛−𝑘 !

Sample Exam #4 An urn contains 10 balls: 4 red and 6 blue. A second urn contains 16 red balls and an unknown number of blue balls. A single ball is drawn from each urn. The probability that both balls are the same color is 0.44. Calculate the number of blue balls in the second urn

Sample Exam #248 Bowl I contains eight red balls and six blue balls. Bowl II is empty. Four balls are selected at random, without replacement, and transferred from bowl I to bowl II. One ball is then selected at random from bowl II. Calculate the conditional probability that two red balls and two blue balls were transferred from bowl I to bowl II, given that the ball selected from bowl II is blue.

Actex, Sec 3 #1, pg 103 A class contains 8 boys and 7 girls. The teacher selects 3 of the children at random and without replacement. Calculate the probability that the number of boys selected exceeds the number of girls selected.

Actex, Sec 3, #3, pg 103 A box contains 4 red balls and 6 white balls. A sample of size 3 is drawn without replacement from the box. What is the probability of obtaining 1 red ball and 2 white balls, given that at least 2 of the balls in the sample are white?

Actex, Sec 3, #5 A box contains 35 gems, of which 10 are real diamonds and 25 are fake diamonds. Gems are randomly taken out of the box, one at a time without replacement. What is the probability that exactly 2 fakes are selected before the second real diamond is selected?