 A coin is tossed and a die is rolled. Construct a tree diagram to list all of the possible outcomes. (4 points)
Example: (head, 4) is one possible outcome.
- The serial number on a dollar bill consists of a letter, 8 digits, and then another letter. How many different serial numbers are possible given the following?
- Letters and digits can be repeated. (5 points)
- Letters and digits cannot be repeated. (5 points)
A softball team of 5 men and 5 women must be arranged in a batting order.
- How many different batting orders are possible? (5 points)
- How many are possible if the pitcher must be the last batter? (5 points)
- How many different batting orders are possible if the odd-numbered batters are men and the even-numbered batters are women? (5 points)
A group of 8 women and 6 men must select a 4 person committee. How many different committees are possible if it must consist of the following?
- any mixture of men and women (5 points)
- two women and two men (5 points)
- a majority of women (5 points)
How many anagrams can be made from the following words? (8 points)
(The anagram does not have to be a real word.)
- CAT
- PAPER
- BANANA
- MISSISSIPPI
Nine athletes are trying out for a team. How many different combinations of athletes could be on the team if it accepts: (8 points)
- 0 people
- 1 person
- 2 people
- 3 people
- 4 people
- 5 people
- 6 people
- 7 people
- 8 people
- 9 people
Handshakes
If every member of a group of people shook right hands with every other member, how many handshakes would have taken place. Figure for group sizes 2, 3, 4, 5, 10, and 100. (One method is to draw a dot for each person and count the number of ways to connect two dots.) (10 points)
Four Weights
Using a balance scale and four weights you must be able to balance any integer load from 1 to 40. How much should each of the four weights weigh? (10 points)
Breakable Billiard Balls
In front of you is a 100 story building. You must determine which is the highest floor you can drop a billiard ball from without it breaking. You have only two billiard balls to use as test objects. If both of them break and you don't know the answer then you have failed at your task. What is the least number of drops needed to be sure you will have determined the breaking point? (10 points)
The Spider and the Ant
There is a block of wood 9" by 9" by 22" with the long edge laying along a north/south direction. An ant is sitting on the north end of the board, halfway up vertically, and 1" from the east edge. A spider is sitting on the south end of the board, halfway up vertically, and 1" from the west edge. The spider can crawl at a rate of 1" per minute. The ant figures that it will take 1+22+8=31 minutes for the spider to reach him so he dozes off for a 30 minute nap. Just as the ant wakes up the spider kills him. By what route did the spider take to get to the ant in 30 minutes? (10 points)
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- BONUS QUESTION: Meetings On The Dot
The hour, minute, and second hands of an ordinary clock cannot coincide exactly except at twelve o'clock. At what other moment in the day are the three hands closest to perfect alignment? (10 points)
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