Combinations Examples

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Combinations Examples Example 1: From a group of 10 books, how many different pairs can you choose to take on your next trip?

Combinations Examples Example 1: From a group of 10 books, how many different pairs can you choose to take on your next trip? C(10, 2) = (10•9)/2 = 45

Combinations Examples Example 2: From a group of 10 books, how many different groups of 3 can you choose to take on your next trip?

Combinations Examples Example 2: From a group of 10 books, how many different groups of 3 can you choose to take on your next trip? C(10, 3) = (10•9•8)/(3!) = 120

Combinations Examples Example 3: From a group of 10 books, how many different groups of 4 can you choose to take on your next trip?

Combinations Examples Example 3: From a group of 10 books, how many different groups of 4 can you choose to take on your next trip? C(10, 4) = (10•9•8•7)/(4!) = 210

Combinations Examples Example 4: From a group of 10 books, how many different groups of 6 can you choose to take on your next trip?

Combinations Examples Example 4: From a group of 10 books, how many different groups of 6 can you choose to take on your next trip? C(10, 6) = (10!)/(4!6!) = 210 (Notice that C(10, 6) = C(10, 4). Why?)

Combinations Examples Example 5: You’ve chosen to buy a large sundae with vanilla ice cream. You can choose 1, 2, or 3 toppings from a total of 7 available toppings. In how many different ways can you top your sundae?

Combinations Examples Example 5: You’ve chosen to buy a large sundae with vanilla ice cream. You can choose 1, 2, or 3 toppings from a total of 7 available toppings. In how many different ways can you top your sundae? 1 Topping: C(7, 1) = 7 2 Toppings: C(7, 2) = 21 3 Toppings: C(7, 3) = 35 Total Possibilities: 7 + 21 + 35 = 63

Combinations Examples Example 6: A standard deck of cards contains 52 cards to include the numbers 2-10, a Jack, Queen, King, and Ace in each of four suits: Hearts and Diamonds are red, Spades and Clubs are black. In how many ways can you get 4 of a kind?

Combinations Examples Example 6: A standard deck of cards contains 52 cards to include the numbers 2-10, a Jack, Queen, King, and Ace in each of four suits: Hearts and Diamonds are red, Spades and Clubs are black. In how many ways can you get 4 of a kind? There are 13 possibilities for face value: The numbers 2-10, a face card, or the Aces. The 5th card could be any of the remaining 48 cards in the deck. 13(48) = 624

Combinations Examples Example 7: A standard deck of cards contains 52 cards to include the numbers 2-10, a Jack, Queen, King, and Ace in each of four suits: Hearts and Diamonds are red, Spades and Clubs are black. In how many ways can you get 3 of a kind?

Combinations Examples Example 7: A standard deck of cards contains 52 cards to include the numbers 2-10, a Jack, Queen, King, and Ace in each of four suits: Hearts and Diamonds are red, Spades and Clubs are black. In how many ways can you get 3 of a kind? There are 13 possible face values, as in the previous example. However, we have more than 1 way to combine each value. For example, there are C(4, 3), or 4 ways to have 3 Aces. Continued… ↘

Combinations Examples Example 7: A standard deck of cards contains 52 cards to include the numbers 2-10, a Jack, Queen, King, and Ace in each of four suits: Hearts and Diamonds are red, Spades and Clubs are black. In how many ways can you get 3 of a kind? Say we have 3 Kings. Neither the 4th nor the 5th card can be a King, because that would give us 4 of a kind. Also, the 4th and 5th cards cannot have the same face value, which would result in a full house. There are C(12, 2) = 66 possible pairings of face values, and each card can be any suit. This gives us 4(13)(66)(42) = 54,912 possibilities.

Combinations Examples Example 8: A flush consists of 5 cards of the same suit. How many flushes are possible in a deck of cards?

Combinations Examples Example 8: A flush consists of 5 cards of the same suit. How many flushes are possible in a deck of cards? There are 4 suits. Within each suit, there are C(13, 5) = 1287 combinations of 5 cards. However, we should exclude the 10 possible straight/royal flushes in each suit. Thus there are 4(1277) = 5108 possible flushes.

Combinations Examples Example 9: A straight flush consists of 5 cards in order and of the same suit, such as 8-9-10-J-Q of Diamonds. How many straight flushes are possible in a deck of cards?

Combinations Examples Example 9: A straight flush consists of 5 cards in order and of the same suit, such as 8-9-10-J-Q of Diamonds. How many straight flushes are possible in a deck of cards? If we arrange the cards from lowest value to highest, then we have 9 starting points: A-9 (A-2-3-4-5 through 9-10-J-Q-K, excluding 10-J-Q-K-A because that is a royal flush). Because we have 4 suits, there are a total of 9(4) = 36 possible straight flushes.

Combinations Examples Example 10: A straight consists of 5 cards in order, such as 8-9-10-J-Q. How many straights are possible in a deck of cards?

Combinations Examples Example 10: A straight consists of 5 cards in order, such as 8-9-10-J-Q. How many straights are possible in a deck of cards? We can arrange the cards again from lowest to highest, starting with A-10, which gives us 10 possible sets of face values. We can have any combination of suits, as long as we don’t have 5 of the same suit—which would be a straight flush. (Continued on next slide)

Combinations Examples Example 10: A straight consists of 5 cards in order, such as 8-9-10-J-Q. How many straights are possible in a deck of cards? Consider the straight 6-7-8-9-10, for example. There are 4 possibilities for the 6, because it can be any suit. Likewise there are 4 possibilities each for the 7, 8, 9, and 10. We don’t want all cards to be the same suit, though, so subtract the 4 possibilities of that (4 hearts, or 4 diamonds, etc.). (Continued on next slide)

Combinations Examples Example 10: A straight consists of 5 cards in order, such as 8-9-10-J-Q. How many straights are possible in a deck of cards? So we have 10 possible groupings of face values, and 45 – 4 = 1020 possible combinations of cards within each of those face value groupings. This gives us a total of 10(1020) = 10,200 straights, not including straight flushes or royal flushes.