Illinois State University Mathematics Department

MAT 305: Combinatorics Topics for K-8 Teachers



Sample Semester Exam Questions

possible solutions


1.a

Replace a, b, c, and d in C(10,8) = C(a,b) + C(c,d) to illustrate Pascal's Formula.

1.b

An individual's telephone number is made up of a 3-digit area code and a 7-digit local number. Until quite recently, the area code could not begin with 0 or 1, the area code was required to have a 0 or 1 as its middle digit, and the local number could not begin with 0 or 1. Under these conditions, how many 10-digit individual numbers were possible?

1.c

A group of 11 people are to be separated into three rooms, A, B, and C, so that five are in room A, two are in room B, and four are in room C. In how many ways can this be done?

1.d

A shelf is to contain 10 books, six indistinguishable paperback books and four indistinguishable hardback books. If the paperback books must be shelved in pairs (that is, exactly and only two paperback books must be adjacent to each other), in how many ways can the 10 books be arranged on an open shelf?

1.e

Use symbol manipulation (algebra) to verify that .

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2.

The following statement is to be proven true by induction:

What is the sum of the first n odd positive integers?

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3.

Twenty-four distinct fair dice are rolled. How many ways are there for thirteen 6s to appear?

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4.

Jan works 20 blocks east and 15 blocks north of her home. All streets from her home to her workplace are laid out in a rectangular grid, and all of them are available for walking. On her walk to work, Jan always stops at Benny's Bakery, located 6 blocks east and 4 blocks north of her home. On her way home from work, Jan always stops at Brink's Bank, located 5 blocks south and 2 blocks west of her workplace. If she walks 35 blocks from home to work and 35 blocks from work to home, how many different round-trip paths are possible for Jan?

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5.

As of today there are 186 major-league baseball players with a batting average from .250 to .299. The batting average must be a three-digit decimal value between .000 and 1.000. Explain why at least four of these players must have the same batting average.

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6.

A certain zookeeper has n cages lined up in a row and has two lions that are indistinguishable from each other. The lions must be placed in separate cages and they may not be placed in adjacent cages.

Write a recurrence relation to describe L(n), the number of ways the two lions can be placed into n cages. Be sure to state any initial conditions of the relationship.

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7.

The Washington Post intends to publish k interviews regarding a recent Supreme Count decision. They interviewed six white females, three black females, seven white males, and four black males. The article must include an interview from at least one person from each of the four categories. For example, if k = 6, the interviews may be from two white females, two black females, one white male, and one black male.

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8.

Fran is a professional singer. She claims to have enough jokes in her repertoire to be able to tell a different set of three jokes in her warm-up act, every night of the year, for at least 50 years. What is the minimum number of jokes she must have?

Note that the set of jokes {A,B,C} is considered one set of jokes, no matter what order Fran tells those three jokes.

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9.

How many ways are there to deal n cards to two persons? It is permissible that the persons may receive unequal numbers of cards.

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10.

Exactly 7 chocolate chips are to be distributed at random into 6 chocolate chip cookies. What is the probability that some cookie has at least 3 chips in it?

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Bonus!

In how many ways can 2n people be divided into n pairs?

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