Basic Combinatorics

Introduction to Combinatorics

Multiplication Principle

If there are m ways for event A to happen and n ways for event B to happen, then there are m*n ways for events A and B to happen.

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If I have 5 shirts and 3 pairs of pants, how many total outfits can I make?

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If I roll a die and toss a coin, how many possible outcomes are there?

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Joni wears a school uniform that consists of a skirt or pants, a white shirt, a blue jacket or sweater, white socks, and black shoes. She has 3 pairs of pants, 3 skirts, 6 white shirts, 2 blue jackets, 2 blue sweaters, 6 pairs of white socks, and 3 pairs of shoes. How many possible outfits could she wear to school?

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How many 3 digit whole numbers are there?

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How many 3 digit whole numbers are there that have an odd tens digit and an even hundreds digit?

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Suppose that I flip a coin 10 times and get tails every time. The probability of that happening is 1/

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Factorials

n! = n*(n-1)*(n-2)*...*3*2*1

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Simplify 6!

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Simplify $\frac{40!}{39!}$ 

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 $\frac{8!}{5!3!}$  Simplify

Permutations

A permutation of a set of objects is a way of arranging the objects in a line. To count the number of permutations, we use the multiplication principle

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How many permutations are there of the set $\left\{a,\ b,\ c,\ d,\ e\right\}$ ? 

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How many ways can I rearrange the letters in ROSE?

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If there are 20 members of a club, how many different ways could they choose a president, a vice president, and a secretary?

Overcounting

• Sometimes it helps to overcount and then subtract or divide out the extras.

• For example, if I have a pile of ropes and I can't tell which rope is which, but I want to know how many, how could I do it?

• Count the ends. If there are 12 rope ends, how many ropes are there?

• 12/2 = 6

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If there are 10 people in a room and they all shake hands with each other, how many handshakes will there be?

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How many numbers less than or equal to 100 are divisible by 2 or 3?

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How many ways can you rearrange the letters of the word ELLZEY? (We are assuming that the E's are indistinguishable as are the L's.)

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How many ways can you rearrange the letters of the word CALCULUS?

Combinations

• Combinations are used when we want to select a number of objects but we don't care about the order in which they are selected.

• For example, if I am choosing members of a committee or students for a class or winners of a raffle, I don't care what order in which the group is chosen in. I only care about who the members of the final group are.

• To calculate the number of ways we can choose these groups, we use combinations, which are an excellent example of overcounting.

Combinations

• Suppose I want to choose a group of 4 students from a class of 10. How could I do it?

• First choose 4 students our of the group of 10, where order DOES matter. How many ways can I do this?

• 10*9*8*7=5040

• Then divide by the number of orders that each group of 4 can be chosen. So what is our final answer?

•  $\frac{5040}{4\cdot3\cdot2\cdot1}=210$  

Combinations

• The general formula for how to choose r objects from a group of n objects where order doesn't matter is  $\ _nC_r=\frac{n!}{\left(n-r\right)!r!}$  

• We say this aloud as "n choose r"

• In our last example, we had 10 students and we chose 4, so  $\ _{10}C_4=\frac{10!}{6!4!}=\frac{10\cdot9\cdot8\cdot7}{4\cdot3\cdot2\cdot1}=10\cdot3\cdot7=210$  

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Compute  $\ _8C_3$ 

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If I choose 3 people at random from a class of 18 to receive bonus points, how many ways could I choose this group?

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In blackjack, each player is dealt two cards to start from a standard 52 card deck. How many different pairs of cards could you receive?

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If your original hand consists of an ace and a face card or a 10, you have blackjack and you win. If you are playing with a standard 52-card deck, how many ways can this happen?

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If you are playing with a standard 52-card deck, what are the chances of getting blackjack?

(Write your answer as a percentage rounded to two decimal places.)

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If the game is played with two 52 card decks instead of one, what are the chances you get blackjack on the first deal?

Enter your answer as a percent rounded to the nearest hundreth.

Challenge Questions

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For the Florida Powerball lottery, you must choose 5 numbers from the numbers 1 to 69 and a powerball number from the numbers 1 to 26. How many different lottery tickets are possible?

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How many different towers can be built with 2 red blocks, 2 blue blocks, and 4 green blocks?

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10 coins are tossed. What is the probability that there are exactly 5 tails?

???/256

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How many ways can you seat 6 people around a table if 3 people insist on sitting together?

Assume that we care about the order people sit, not the specific chair in which they sit.

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How many different ways are there to buy 8 fruits that consist of bananas, apples and pears?

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Six 6-sided dice are rolled. How many outcomes are there where the sum of the dice is 10?

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My friend and I both choose 5 books from a reading list of 10 books to read over the summer. What is the probability that we have exactly 2 books in common?

Answer: ???/63