CCE 235 Handout 23

CSCE 235

Handout 23: Questions in Combinatorics

Assigned March 29, 2004

These questions were generated by students during an exercise on problem recognition on March 26, 2004. First, each student was asked to come up with a question in 5 minutes. Then, each student passed his or her question to another student to be rated 1-10 (10 being very difficult, and 1 being very easy). The rating took 3 minutes. The process was repeated two more times. As a result, each question was rated by three different students. An asterisk (*) indicates that the instructor thinks that the question is more difficult than rated. An exclamation point (!) indicates that the instructor thinks that the question is easier than rated. A question mark (?) indicates that the question is confusing to the instructor.

Score	Question
30!	Booth’s algorithm is a method of multiplying binary numbers, in order for it to save time over traditional multiplication, there must be strings of 1s with at least 3 in a row inside the number. Given a number smaller than , how many possibilities have at least 1 string of 3 1s in a row or more?
30!	How many ways to distribute 10 different books to three people A, B, and C such that one person has 4 books and two people each has 3 books?
30*	If 3 ships fired 10 cannons each, and each cannon fired 5 cannonballs of sizes A, B, C, D, and E with A being the largest and E being the smallest, how many ways could a target be struck as long as one ship fired the A-size cannonball first, one ship fired the A-size cannonball last, and the remaining ship fired the C-size cannonball first and the B-size cannonball last?
26.5!!	Six different people sit down to eat. They are people A-F giving their credit cards 1-6, both respectively. How many ways can each person, A-F, not get their credit cards, 1-6, in the same order that they paid with?
26!	How many ways to form a password for an e-mail account which consists of 16 characters or numbers? The characters are case sensitive.
25!	We have a string of 15 letters from the English alphabet. How many different strings contain “ABC” and “XYZ” where all of the letters in the string are distinct?
25!	How many different ID#s could you have if each ID# started with 3 letters (which could only be used once … i.e., ABC or DST, but not AAB) and ending with 7 numbers which could be repeated?
24	Consider Nebraska license plates only: LLL###. How many ways are possible such that a Nebraska license plate (a) has the string “NU”? (b) has the string “UNL” or “NU” or “NEB” or “UN1”?
22.5?	In a tribe of 35 people, there is a need to have a new leader and a new witch. The leader has to be chosen from only a group of 10 officials. The witch has to be chosen from a group of 7 candidates. (a) how many different ways can we have a leader in the tribe? (b) How many ways can we have a witch in the tribe? (c) How many ways could we have a leader and a witch if there was no selection to be made (of officials or candidates) and if they were chosen only from the tribe?
21.5!	A German Shepherd, a Cocker Spaniard, a pig, and a cat are running around the house. How many different ways can that happen? If the cat comes first, how many different ways can this happen?
20	How many ways can you pick 3 US senators if you cannot pick more than 1 from any state?
20	Driver’s license number in Rhode Island are 18 characters long, they are made up of 10 non-distinct letters and 8 distinct numbers. Preceding the driver’s license number is 1 number or character. How many combinations are there for driver’s license numbers in Rhode Island?
17.5	There are 50 people who want to go on a bus tour (a) How many ways can you select 40 people to go? (b) How many ways can you select 40 people if you have to make a seating chart? (c) Two of the people, Matt and Jody, either have to both go or both not go. Now how many ways can you select 40 people to go without a seating chart?
16*	7 letters, A-G. (a) no repetitions: # of ways to arrange so that “FG” are together, and to the right of “D” (b) # of substrings with “AC” and “CD” (c) # of strings with substrings “DE”, starting with “G”
16*	Jack is bowling. His ball gets stuck in the ball return. So he keeps using different balls. He uses 23 bowling balls: 11 black, 3 blue, 4 red, 3 yellow, 1 pink, and 1 green (each color bowling ball is identical). After the 23^rd ball, all the balls dislodge and come out in a random order plus one pin. How many combinations are possible (including the pin)?
12	How many ways can you re-arrange a word “ISSUE”?
10.5*	Given the letters ABCDEFG. How many 3-letter strings can be made? How many 3-letter strings without the substring AB can be made? How many possibilities are there to re-arrange the letters? Is it possible to have a permutation that includes both substrings ABC and BDE?
8	How many ways are there to place 5 identical CDs into 5 spindles?

Question by the instructor:

For a placement exam, the Department is looking for placing 3 students in CS1 and 4 students in CS2. There are 10 students taking the placement exam: 5 male and 5 female students. How many ways are there if

(a) All 3 students placed in CS1 are male?

(b) All 3 students placed in CS1 are male and all 4 students placed in CS2 are female?

(d) All females are placed?