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12.2 Combinations and The Binomial Theorem

Learn about combinations, the binomial theorem, and how to calculate the number of ways certain events can occur. Includes examples and explanations.

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12.2 Combinations and The Binomial Theorem

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  1. 12.2 Combinations and The Binomial Theorem Algebra 2

  2. Using Combinations • Combination: unordered groupings (order does not matter)

  3. Combinations Of n Objects Taken r At A Time • The number of combinations of r objects taken from a group of n distinct objects is denoted by nCr and is given by:

  4. Combinations • When finding the number of ways both events A and B occur, you multiply. (uses words like exactly) • When finding the number of ways that events A or B can occur, you add. (uses words like at most or at least)

  5. Example: • Use a standard deck of 52 cards. • If the order is not important, how many different 7-card hands are possible? • How many of these hands have all 7 cards of the same suit?

  6. Example: • You are taking a vacation. You can visit as many as 5 different cities and 7 different attractions. • Suppose you want to visit exactly 3 different cities and 4 different attractions. How many trips are possible.? • Suppose you want to visit at least 8 locations (cities or attractions). How many different types of trips are possible?

  7. Example: • In a standard deck of 52 cards. How many possible 5-card hands contain exactly 3 kings.

  8. Example: • From a group of 20 volunteers, you are choosing at least 18 to be peer counselors. In how many different ways can this be done?

  9. Combinations: • Sometimes when problems contain words like “at least” or “at most” the faster way to solve is by subtracting out the possibilities you don’t want vs. adding the possibility you do want.

  10. Example: • A restaurant offers 6 salad toppings. On a deluxe salad, you can have up to 4 toppings. How many different combinations of toppings can you have?

  11. Example: • Every committee of the school council must contain at least 1 senior. If there are 7 seniors on the school council, how many different combinations of seniors can be assigned to a committee?

  12. Using the Binomial Theorem • Pascal’s Triangle: the arrangement of nCr in a triangular pattern. Pascal’s triangle has many different patterns and properties. (See pg. 710)

  13. The Binomial Theorem • The binomial expansion of (a+b)n for any positive integer n is:

  14. Examples: • Expand (a+2b3)4. • Expand (3x+y)4 • Expand (x – 5)4. • Expand (2x – y2)3. • Expand (a – 2b3)3.

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