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Counting Techniques

Counting Techniques. How many ways could…?. The Addition Principle. There are 9 boys and 8 girls in the student council at Hermitage High School. How many ways could a single student be selected to hold a single office? There are 9+8=17 ways of making the selection. The Addition Principle.

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Counting Techniques

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  1. Counting Techniques How many ways could…?

  2. The Addition Principle • There are 9 boys and 8 girls in the student council at Hermitage High School. How many ways could a single student be selected to hold a single office? • There are • 9+8=17 ways of making the selection

  3. The Addition Principle • The events of selecting a boy and selecting a girl are called Mutually Exclusiveor Disjoint. This is because a person cannot be both a boy and a girl.

  4. Fundamental Multiplication Principle • How many ways can a die and a coin fall together? • First make two Blanks • _ _ • Thenwrite the number of possibilities for the die and the coin on the blanks. • 62 • Now multiply those numbers together to get 12.

  5. Fundamental Multiplication Principle • Think back to the problem with Hermitage’s student council. If both one boy and one girl are to be selected to hold two different offices on the council, how many ways can the offices be filled? • Remember there are 9 boys and 8 girls and to make your two blanks!

  6. Permutations • Suppose, from the 17 students in the student council, how many ways can you choose a president, secretary, and treasurer? One person cannot hold more than one position. • If the president is elected first, how many options are there? • 17 • How many options are left for the secretary if that position is elected next? • 16

  7. Permutations • How many options are there for the treasurer if that position is chosen third? • 15 • Now use the multiplication principle to find your final answer.

  8. Permutations • If there are 5 students waiting to get in line to get on a bus, how many orders can the first two students get on the bus? • There are 20 different orders for the first two students to get on the bus.

  9. Permutations • The formula for a permutation is… • nPr = • 5 students, the first two would be 5P2 • Note that 6! represents • TI-84 press the number for n, math, PRB, the second one on the list, then the number for r.

  10. Combinations • A Combinationis used to describe a selection of several objects where the order they were chosen does not matter. • Think of the first two students entering the bus. If you only need to know who those first two students were and not the order. For example Brian then Darren would be the same as Darren then Brian.

  11. Combinations • We knew there were 20 orders the first two student could enter the bus. • For each pair of students there were two orders. • How many combinations of the first two students can be made?

  12. Combinations • There are 10 combinations of the first two students on the bus. • nCr= • 5C2=

  13. Review • Permutations: order matters • Combinations: order does not, think groups • If there are 8 students and 8 tests versions, how many ways can the tests be given out? • 8!=40,320 • How many two letter “words” can be made from the word FLOWERS? • 7P2=42 • 9 boys and 8 girls in student council. How many four person comities can consist of 3 boys and 1 girl? • 9C3*8C1=84*8=672

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