##### ### Final Examination: Individual Portion ### Combinatorics Topics For K-8 Teachers MAT 305 Summer 1997 #### Roger Day day@math.ilstu.edu

1.

In a utility drawer in Tim Toolman's workshop are several pairs of gloves. Each pair is identical except for color. There are 3 pairs of red gloves, 4 pairs of blue gloves, and 5 pairs of white gloves. During a blackout, Tim reaches into the drawer without being able to distinguish among the gloves.

(a) How many single gloves must Tim bring out to assure that he has a pair of gloves to wear, with no regard for color?

(b) How many single gloves must Tim bring out to assure that he has a pair of gloves to wear, this time a pair of the same colored gloves?

look at possible solution

2.

Five single newborns were arranged in a line of five cribs in the maternity ward of a local hospital. A new floor nurse had gotten the babies' name tags mixed up. Three babies had their correct name tags and two got switched.

(a) In how many ways could the name-tag switch have occurred with this group of newborns?

(b) Suppose we know there are three female newborns and two male newborns, and that one female and one male had their name tags switched. How many ways could this occur?

(c) In a maternity ward with n cribs lined up, in how many ways could all the name tags be switched, that is, none of the cribs had the correct tag?

look at possible solution

3.

Sam's cafe has 10 counter seats in a straight line. Each morning the same three grouchy people come to Sam's for coffee and refuse to speak to each other until finishing one fresh cup of coffee.

(a) How many different ways can these three grouchy customers occupy the ten seats so that no two are sitting adjacent?

(b) Repeat question (a) for k grouchy people at a counter with n seats. Explain any restrictions on the relationship between k and n.

look at possible solution

4.

 Start at the center letter M and move outward. Each succeeding move should be up, down, left, or right. How many different paths spell the word MATH?

look at possible solution

5.

Justin was preparing a card trick that used the nine numbered cards (2,3,4,5,6,7,8,9,10) from one suit of a standard 52-card deck. The nine cards are in a hat and Justin will draw one card at a time from the hat. Justin cannot see into the hat and he does not put cards back in after drawing.

(a) How many different ways can Justin draw six cards from the hat?

(b) Justin's friend Melanie also is practicing card tricks. She uses the same nine cards as Justin and draws one card at a time from a hat. Melanie, however, records the numerical value of the drawn card, returns it to the hat, mixes up the cards, and draws again. How many ways can Melanie draw four cards whose numerical values sum to 16?

look at possible solution

6.

A public opinion poll of 1200 adults shows that

• 675 are married
• 682 range from 20- to 30 years old
• 684 are female
• 195 are married and are from 20 to 30 years old
• 467 are married females
• 318 are females ranging in age from 20 to 30 years
• 165 are married females ranging in age from 20 to 30 years

(a) In this group of 1200 adults, how many were unmarried males not in the 20- to 30-year old age?

(b) In this group of 1200 adults, how many married males range in age from 20 to 30 years?

(c) How many unmarried males were questioned?

look at possible solution