##### ### Test #3 possible solutions ### Combinatorics Topics For K-8 Teachers MAT 305 Summer 1997 #### Roger Day day@ilstu.edu

1.

Consider the expansion of the multinomial (m + n + p + s + v)^11.

a) Determine the number of uncollected terms in the expansion of this multinomial.

b) Determine the number of collected terms in the expansion of this multinomial.

c) Determine the value of K for the collected term Km^4n^2pv^4

BONUS! A collected term in the expansion of the multinomial shown above has coefficient 11. How many collected terms will have that coefficient?

look at possible solution

2.

Consider the word microfibrillar.

a) How many unique arrangements are there for the letters in this word?

b) How many ways are there to arrange the letters in the word and keep the vowels from being adjacent to one another?

BONUS! Write down a legitimate English-language word that fits all the following criteria. (Webster's Unabridged, in STV 313, will be used to certify words):

i) The word has no fewer that 6 letters.
ii) At least one of the letters appears more than once.
iii) There are no less that 16,480 unique arrangements of the letters in your word.

DOUBLE BONUS!! Determine the maximum number of letters for a word that meets all the following conditions:

i) There are at least 4 unique letters in the word.
ii) At least 2 different letters in the word repeat.
iii) There are fewer than 8910 unique arrangements of the letters in the word.

look at possible solution

3.

Joannie Jorgensen volunteers at the downtown Senior Citizen Center. She's responsible for keeping records and reporting statistics that describe each day's luncheon clientele. She records the following information for each diner:

• the gender of the diner
• whether the diner is a local resident or from out of town
• whether the diner is a member of AARP
• whether the diner has previously eaten at the Center

Last Friday, Joannie reported information about the day's luncheon diners. Here's the beginning of her report:

DINER REPORT FOR FRIDAY
filed by Joannie Jorgensen

57 diners in all:

17 male/local resident/AARP members/returning diners
16 female/local resident/AARP members/returning diners
0 male/local resident/AARP members/first-time diners
3 female/local resident/AARP members/first-time diners
. . . and so on . . .

a) How many different descriptions are possible for diners according to the records kept by Joannie?

b) Yesterday there were 38 diners. How many different reports, similar to the one above, could have been filed by Joannie yesterday, given that some descriptions may have had no diners, such as the third entry in the example?

look at possible solution

4.

Two octopi took part in a friendly tentacle-to-tentacle wrestling match. Each managed to pin 4 of its opponent's tentacles with 4 of its own. In how many ways could the match have taken place?

look at possible solution

5.

5. Fifty tickets, numbered consecutively from 1 to 50, are placed in a #2 mayonnaise jar on Funk & Wagnall's porch. Two tickets are drawn from the jar. The order the two tickets are drawn is not important.

a) Verify that there are 1225 ways for this to occur. Explain your verification.

b) Of the 1225 possible pairs of tickets, how many pairs show two numbers whose difference is 10 or less?

look at possible solution

6.

Here is a problem situation and the work submitted by a student:

 A shelf is to contain nine different books, six different paperback books and three different hardback books. If the paperback books must be shelved in pairs (that is, exactly two paperback books must be adjacent to each other), how many ways can the nine books be shelved? First, place the hardback books so 2 paperbacks can go between them. There are 3! ways this can be done. To complete the arrangement, there are 8 places left to place 6 paperback books. Since the books are different and order matters, we have a permutation, P(8,6). The number of arrangements of the two types of books must be multiplied together since the total arrangement depends on the placement of each type. Solution: 3!*P(8,6)

Analyze the student's work. Is it correct? If so, identify the key step or steps in the student's solution. If the work is incorrect, identify the error or errors in the student's work and suggest what should have been done.

look at possible solution