Binomial Theorem

Binomial Theorem

Venn Diagrams

Mixed Combinatorics A

Mixed Combinatorics B

Mixed Combinatorics C

100

Find the 2nd term in the 5th row of Pascal's triangle.

5

100

Draw a 2 circle Venn Diagram representing "not P AND Q"

everything that is in Q only

100

Three identical door prizes are to be given to three lucky people in a crowd of 100. In how many ways can this be done?

100! / (100-3)! 3!

100

How many 4-digit numbers contain no 1's?

8*9*9*9

100

A student must take four final exams, scheduled by a computer, during the morning and afternoon testing periods on Monday through Friday of exam week. If the order of the student's four exams is important, in how many ways can a computer schedule the exams?

10 P 4 = 10! / ((10-4)!(4!))

200

Expand (x + y) ^3 using the binomial theorem.

x ^3 + 3x^2 y ^1 + 3x y^2 + y^3

200

In a survey about breakfast foods of 60 high school students, 25 liked cereal, 20 liked scrambled eggs, and 30 said they didn't like either. How many like cereal but not scrambled eggs?

10

200

In how many ways can the letters of the word RADISH be arranged?

6!

200

In how many ways can the letters of the word TOMATO be arranged?

6! / 2!

200

There are 3 roads from town A to town B, 5 roads from town B to town C, and 4 roads from town C to town D. How many ways are there to go from A to D via B and C?

3 x 5 x 4

300

In the expansion of (x - 2 )^15, find the coefficient of the term containing x^13.

C (15,2) *x^13 * (-2)^2 = 15!/ (15-2!2!)* 4 = (15*14*2)is the coefficient

300

Of the 320 children attending a matinee at a movie theater, 190 buy popcorn, 245 buy something to drink, and 151 buy both. How many children buy neither popcorn nor something to drink?

P U D = 190 + 245 - 151 = 284
not ( P U D) = 320 -284 = 36

300

A teacher must pick 3 high school students from a class of 30 to prepare and serve food at the junior high school picnic. How many choices are possible?

30! / (30- 3)!

300

The locks on the gym lockers have dials with numbers from 0 to 39. Each locker combination consists of 3 numbers. Pythagoras remembers that his numbers are 8, 15, and 17, but he can't remember in which order they appear. If he can try one possibility every 10 seconds, what is the maximum amount of time that is would take him to find the right combination?

3! x 10 second = 60 seconds

300

In how many ways can 8 jackets of different styles by hung on a circular rack?

8! / 8 = 7!

400

In the expansion of (x^2 - 2)^15, find the coefficient of x^12.

C(15,9) * (x^2)^6 * (-2)^9

400

Draw and shade a 3 circle Venn Diagram for
"(B union C) intersect (B intersect A)"

must be in only B

400

If you have a penny, a nickel and a dime, how many different sums of money can you make using one or more of these coins?

3C3 + 3C2 + 3C1
= 1 + 3 + 3
= 7

400

Five girls and five boys stand in a line. How many arrangements are possible if all the boys and girls stand alternately?

Boy first: 5*5*4*4*3*3*2*2*1*1 = 5!*5!
Girl first: 5*5*4*4*3*3*2*2*1*1 = 5!*5!
2*5!*5!=28800

400

How many 5 - digit numbers contain at least one 3?

90000 - 8*9*9*9*9=37512

500

Find the fourth term in the expansion of (x^2-1/x^2)^6

What is -20?

500

Create a 3 circle Venn Diagram and fill in the regions as follows: 90 people were asked about baseball, basketball, and football. 5 people liked all three. 15 people liked baseball and basketball. 25 people liked football and basketball. 10 people like baseball and football. 25 people liked baseball. 40 people liked football. 55 people liked basketball.
_____ people liked none of the the three sports.

What is 15?

500

In the World Series, two teams, A and B, play each other until one team has won 5 games. For example, the "word" ABBAAAA represents a 7-game series in which team A wins games 1, 4, 5, 6 and 7. What is the number of possible different 7-game series won by team A?

6! / (4! 2!) = 15

500

A jar contains 3 black marbles and 4 white marbles. Ms. W and Mr. P takes turns picking a marble from the jar (without putting it back). If Ms. W goes first, how many ways can she win? Show them.

Win on 1st turn: B
Win on 2nd turn: WWB
Win on 3rd turn: WWWWB

500

In the tennis championship between players A and B at Wimbledon, the first player to win 3 sets is a champion. Find the number of different ways for player A to win the championship.

WWW arranged in 3!/3! ways = 1
WWWL arranged in 4!/3! ways = 4
WWWLL arranged in 5!/ (3!2!) ways = 10
total = 15

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