Counting

1
Counting

• If you go to Baskin-Robbins (31 flavors) and make
  a triple-scoop ice-cream cone,
• How many different arrangements can you create
  (if you allow repeats)?
• If you dont allow repeats?

31.31.31 313 29791
31.30.29 26970
2
Counting

• You need to visit 5 cities. How many possible
  different orderings of visiting the cities (no
  repeats) are there?

3
Permutations and Combinations

• These deal with drawing items without replacement
• Are different orderings counted separately?
• In other words, is ABC considered different than
  BAC?
• If YES, were talking about Permutations
  (arrangements)
• If NO, were talking about Combinations

4
Permutations

• How many different 4-number PIN numbers are
  possible if no two numbers are the same?

5
Permutations

• The number of permutations (arrangements or
  sequences) of r items selected from n available
  items (without replacement) is

6
Permutations

• Suppose you have to pick 5 photos out of 15 for a
  magazine layout. Ordering matters (different
  layouts should be considered as separate). How
  many possible layouts are there?

7
Combinations

• The number of combinations of r items selected
  from n different items (without replacement) is

The extra term in the denominator makes it so
different arrangements arent counted separately
(ABC is considered equivalent to CBA)
8
Combinations

• Suppose you have to pick 4 of your 12 friends to
  take on a free vacation you won. How many
  possibilities are there?

For comparisons sake, the number of permutations
would be 11880
9
Combinations for Probability

• Suppose you have to pick 4 of your 12 friends to
  take on a free vacation you won. You decide to
  choose by pulling names from a hat. What is the
  probability that Ann, Betty, Carlos, and Dean are
  chosen?
• Thats one choice out of all possibilities, so

10
Homework

• 3.7 1, 3, 5, 9, 21, 25