Combinations

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Combinations

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COUNTING SUBSETS

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A SET is an unordered collection of things.
RAT, ART, and TAR are each the same set of letters.
A SUBSET is a set that is part of a set.
RATE, EARI, ARTI, IATE and TERI are each different subsets of IRATE.
But RARE and TREAT are not subsets because sets have no duplicates.
The number of things in a set is called its SET SIZE.
The IRATE set has 5 letters; its size is 5.
Subsets of IRATE may have 5, 4, 3, 2, 1, or 0 set size.
Let n be the set size, and k be the subset size.
A COMBINATION is a subset of things where the order does not matter.
The notation _nC_k means the number of subsets of k things out of n things.
In other words, _nC_k is the number of combinations of n things taken k at a time.
So RATE is a 4 letter combination of IRATE,
and TEAR is the same combination.
The number of subsets of 4 letters from IRATE is ₅C₄ = 5.

QUESTIONS: SET I
What extra step(s) must you do to convert the number of arrangements of k things out of n things into the number of subsets of k things out of n things? List all the possible subsets of smaller sets first until you detect a pattern. Fill out the table below. Then write a formula using factorials based on techniques of counting that you have used before.

How many k letter subsets can you make from n letters? Write a rule that counts the subsets of k things taken from n different things. Explain your rule. Why does it work?
Write a formula for your rule where n is any number of letters, and k is the number of letters that you select from the n letters.
How is your rule similar to the rule for counting arrangements of subsets?
How does your rule differ from the rule for counting arrangements of subsets? State the obvious.
What extra operation must you do to convert a permutation into a combination?

Use the table above to answer the following questions about Quincy pizzas. Quincy pizzas are cheap but only offer a maximum of 6 toppings. How many Quincy pizzas can you make with exactly...

0 toppings? (This is the plain cheese pizza)
1 topping? (Pepperoni)
2 toppings? (Pepperoni, Sausage)
3 toppings? (Pepperoni, Sausage, Mushrooms)
4 toppings? (Pepperoni, Sausage, Mushrooms, Peppers)
5 toppings? (Pepperoni, Sausage, Mushrooms, Peppers, Olives)
6 toppings? (Quincy's Deluxe Supreme with Anchovies)

How many pizzas in all can you order from Quincy's?

QUESTIONS: SET II

Assume that all n objects are different.
None of the k objects is repeated.
Order of the k objects is NOT counted.

When bowling, it does not matter what order of pins we knock down, or what the order of pins are that still stand. How many ways are there to leave 3 bowling pins out of 10 standing? You could express this question as find ₁₀C₃.
There are only 18 computers in the computer lab. Fifteen students have already enrolled in Math 1013, but 5 more late registering students have requested enrollment also. Only 3 of these students can be enrolled. How many sets of three students may be selected to fill the three desks? List all the sets of three students who may be selected to fill the three desks.
List all the different letters in your first and last names.
1. How many letters are in your list?
2. How many ways can you arrange these letters? Show how.
3. How many ways can you pick 4 letters to arrange from these letters?
4. Show how you derived your answer.
5. How many ways can you pick sets of 4 letters from these letters?
6. Show how you derived your answer.
Since almost every country in the world is represented at the United Nations, it is not an exaggeration to say that the United Nation is a microcosm of the world. The Organization uses six official languages in its intergovernmental meetings and documents: Arabic, Chinese, English, French, Russian and Spanish. How many official translators are needed at the United Nations?
Nikki bought a new car with a radio that has six buttons for AM stations and five buttons for FM stations. There are twenty-four AM stations and fifteen FM stations from which she can choose.
1. How many ways can Nikki program her radio for AM stations?
2. How many ways can Nikki program her radio for FM stations?
3. How many ways can Nikki program her radio for all eleven stations?
A poker hand consists of 5 cards from a deck of 52 cards.

How many poker hands...
1. are possible in all?
2. contain 4 cards of the same rank?
3. contain 4 aces?
4. are flushes where all cards are of the same suit?
5. are full houses where 2 cards are of one rank and 3 cards are of another?
6. are full houses where 2 cards are of rank 8 and 3 cards are Aces?
7. contain 3 cards of the same rank, but the other 2 cards are not of the same rank? In other words, 3 of a kind, but not a full house or 4 of a kind.
The Lucky 4 game card contains 16 Smiley faces. Six of these Smileys are winners. Find four of the winners, and you win $100. But you can only choose four.
1. How many different ways can you choose four Smileys?
2. How many ways can you choose four winners?
3. If you can find all six Smileys, you win $500. Having found the four winning Smileys, how many ways can you choose the other two?
The Lucky 7 game card contains 9 Smiley faces. Five of these Smileys cover up a 7. If you can get three 7's in a row vertically, horizontally, or diagonally, you win $2.
1. How many different ways can the 7's be placed on the card?
2. How many ways can 3 of those 5 7's be placed on row 1?
3. How many ways can 3 of those 5 7's be placed on column 1?
4. How many ways can 3 of those 5 7's be placed on one diagonal?
5. How many ways can you win?
6. How many ways can you lose?
7. Play this game 10 times and count how many times you win.
8. This game places the 7's randomly.
  Do you think that the Tennessee Lottery Lucky 7 game also places the 7's randomly?
QUESTIONS: SET III
Which of the questions 1 thru 8 above is illustrated by the mathlet below.
- Using the mathlet above, illustrate each of the questions below.
- Then write the formula using factorials that will calculate the answer.
- Then calculate the answer.
How many ways can 12 people can shake hands?
How many tennis matches are possible with 20 players?
There are seven visible objects in the solar system that you can see without a telescope: the Sun, the Moon, Mercury, Venus, Mars, Jupiter, and Saturn. How many ways may any two of these solar system objects align so that only one of them is seen? This phenomenon is called a conjunction. Some conjunctions are eclipses.
How many ways can you be dealt two hole cards in Texas Holdem if Ace of Hearts and Ace of Spades is the same as Ace of Spades and Ace of Hearts? In other words, the order you are dealt your hole cards does not matter.
How many different flavors of ice cream must you buy to make thirteen different double scoop cones? (Vanilla & Chocolate is the same as Chocolate & Vanilla.) Careful, the answer is not 13.
How many blocks are required to build a staircase with 19 steps? Careful, this one is a bit tricky...

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Last Modified 10/23/08 8:45 AM

Combinations

QUESTIONS: SET I

QUESTIONS: SET II

QUESTIONS: SET III