Combinatorics

Factorials and Pascal's Triangle

100

What is the 5th line of Pascal's triangle?

1 5 10 10 5 1

200

Evaluate 6!/3!

6!/3! = 4x5x6 = 120

300

How can you create Pascal's Triangle using combinations?

Each number is nCr.  The first row is 0C0.  Then 1C0 and 1C1.  Then 2C0, 2C1, 2C2.  Then 3C0, 3C1, 3C2, 3C3, etc.

400

What is the full formula for nPr?

(n!)/((n-r)!)

500

How many numbers are there in the (n+4)th row of Pascal's triangle?

n+5

Binomial Theorem and Expansion

100

Evaluate 5 C 3 (5 choose 3)

5 C 3 = 10

200

Evaluate 8 C 5 (8 choose 5)

8 C 5 = 56

300

Write the expansion for (x + y)^5

1x^5 y^0 + 5x^4 y^1 + 10x^3 y^2 + 10x^2 y^3 + 5x^1 y^4 + 1x^0 y^5

400

What is the coefficient for the term x^7 y^2 in the expansion of (x + y)^9 ?

9! / (7!2!) = 36

500

Evaluate n C (n-1) (n choose n-1)

n

Permutations

100

Set up but do not evaluate: In how many different ways can 9 floats line up for a parade?

9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

200

Set up but do not evaluate: A customer can choose one of 4 different salad mixes, one of 8 different toppings, and one of 5 dressings. How many different salads can s/he create?

8 x 4 x 5 or 160 different salads

300

Set up but do not evaluate: In how many different ways can you answer a 14 question true or false quiz?

2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 or 2^14

400

Set up but do not evaluate: In a certain state, the license plate is made up of 3 letters followed by 4 numbers. If the letters may not be used more than once, and the first number must be greater than 5, but any number can be reused, how many different license plates are possible?

26 x 25 x 24 x 4 x 10 x 10 x 10

500

A group of 25 people are completing in an essay competition.  The top 5 winners each win a monetary prize of $500, $400, $300, $200, or $100 based on what place they come in.  Find the number of possibilities in this scenario.

6,375,600

Combinations and Permutations

100

Set up but do not evaluate. The game euchre uses only 24 cards. How many 5-card hands can be made in the game?

24 C 5

200

Set up but do not evaluate: Your school yearbook has an editor-in-chief and an assistant editor-in-chief. If the yearbook staff has 15 members, in how many different ways can these positions be filled?

15 x 14

300

Set up but do not evaluate: A video store receives 11 new movie posters. The store manager only has wall space for 4 posters. In how many ways can the manager choose which posters s/he will hang?

11 C 4

400

Set up but do not evaluate: A relay race has 4 runners who run different parts of the race. There are 16 students on your track team. In how many different ways can your coach select students to compete in the race?

16 x 15 x 14 x 13

500

Set up but do not evaluate: A box contains 12 black and 8 green marbles. In how many different ways can you choose 3 black marbles and 2 green marbles?

(12 C 3 ) x (8 C 2)

Probability

100

You roll two 5-sided dice. What is the probability that their product is a multiple of 3?

9/25 (Hint: make a table of products)

200

You roll a fair six-sided die.  The die shows an odd number or a number greater than four.  Find the probability.

2/3

300

P(A) = 3/10.   P(B┃A) = 7/20.  Find P(A and B)?

21/200

400

In a bag of marbles, there are 5 red marbles labeled 1 through 5 and 8 blue marbles labeled 1 through 8. What is the probability that you draw a red marble a an even numbered marble?

5/13 + 6/13 - 2/13 = 9/13

500

In a bag of marbles, there are 5 red marbles labeled 1 through 5 and 8 blue marbles labeled 1 through 8. You draw two marbles without replacement. What is the probability that you draw a blue, odd-numbered marble first and that you draw a multiple of 4 next?

(4/13) x (3/12) = (4/13) x (1/4) = 1/13