Use Combinations and the Binomial Theorem

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Use Combinations and the Binomial Theorem Notes 10.2 Use Combinations and the Binomial Theorem

Order of events or outcomes DOES MATTER!!! Combination: Order of events or outcomes IS NOT IMPORTANT!!! Permutation: Order of events or outcomes DOES MATTER!!!

Combinations nCr Examples: 8C3 10C6 7C2 14C5

Pascal’s Triangle!

Finding Combinations A standard deck of 52 playing cards has 4 suits with 13 different cards in each suit. If the order in which the cards are dealt is not important, how many different 5-card hands are possible? In how many 5-card hands are all 5 cards of the same color? 4 queens and 1 other card 5 non-face cards

Solutions: The number of ways to choose 5 cards from a deck of 52 cards is: 52C5 For all 5 cards to be the same color, you need to choose 1 of the 2 colors and then 5 of the 26 cards in that color. So, the number of possible hands is: 2C1 · 26C5

Multiple Events When finding the number of ways both event A AND event B can occur, you MULTIPLY. When finding the number of ways that event A OR event B can occur, you ADD. At least one heart At least one Ace

William Shakespeare wrote 38 plays that can be divided into three genres. Of the 38 plays, 18 are comedies, 10 are histories, and 10 are tragedies. How many different sets of exactly 2 comedies and 1 tragedy can you read? How many different sets of at most 3 plays can you read?

During the school year, the girls basketball team is scheduled to play 12 home games. You want to attend at least 3 of the games. How many different combinations of games can you attend?

Binomial Expansion There is a relationship between the powers of binomials and combinations. Example: 4C0 + 4C1 + 4C2 + 4C3 + 4C4

Binomial Theorem: Whenever a binomial has a POSITIVE EXPONENT (n is positive), the binomial expansion of (a + b)n is: nC0anb0 + nC1an-1b1+nC2an-2b2+…+nCna0bn This sequence stops at n (the power).

Use the binomial theorem to write the binomial expansion.

Use the binomial theorem to write the binomial expansion.

Use the binomial theorem to write the binomial expansion.

Use the binomial theorem to write the binomial expansion. *** IF the binomial is a difference (subtraction), use the change-change method to make it a sum.

Use the binomial theorem to write the binomial expansion. *** IF the binomial is a difference (subtraction), use the change-change method to make it a sum.

Use the binomial theorem to write the binomial expansion. *** IF the binomial is a difference (subtraction), use the change-change method to make it a sum.

Homework: P 694 3-10, 13-18, 24-34, 38-40