Illinois State University Mathematics Department

MAT 305: Combinatorics Topics for K-8 Teachers

Spring 1999
6:00 - 8:50 pm Tuesday STV 332
Dr. Roger Day (

Test #2
Possible Solutions

Please write your solutions on one side only of each piece of paper you use, and please begin each solution on a new sheet of paper. You may use factorial notation as well as combination and permutation notation where appropriate (i.e., there is no need to expand 24!).

You are to work alone on this test. You may not use anyone else's work nor may you refer to any materials as you complete the test. You may ask me questions.

Evaluation Criteria

You may earn up to 10 points on each of questions 1 through 6. For each question:

  • 6 points count toward a correct solution to the problem. I will evaluate the mathematics you use:
    • Is it accurate and appropriate?
    • Have you provided adequate justification?
  • 4 points count toward how you express your solution. I will evaluate how you communicate your results:
    • Is your solution clear and complete?
    • Have you expressed logical connections among components of your solution?

Note that on Question 1, no commentary is required.


Respond to each of these questions by placing your solution in the blank. While you may show steps leading to your solution, you do not need to generate written explanations for questions (a) through (e) of question (1).

a. Express C(10,4) in terms of P(10,6).

b. How many distinct arrangements exist for the letters in the word ATTESTANT?

c. In the expansion of (m + n + p)^10, determine the number of:

(i) uncollected terms

(ii) collected terms

d. Express C(n,k) + C(n,k + 1) as a single combination.

e. Let S(k) represent the sum of the elements in row k of Pascal's Triangle. For example, S(0) = 1 and S(1) = 2. Express S(12) - S(10) in terms of S(10).



Members of a mathematics department will take a one-day course on using the TI-89 calculator in collegiate mathematics instruction. All 36 instructors in the department must take the course and can take the course only once. The course will be offered on the Monday, Tuesday, Wednesday, and Thursday immediately after spring semester graduation. Faculty must sign-up in advance for the day they wish to complete the course.

In the following questions, we are not concerned about who among the faculty members take the course on a certain day, but about how many take the course each day.

a. If there are no restrictions about how many take the course on any one day, how many ways exist for the department to sign up for the courses?

b. If at least one faculty member must sign-up for each of the days the course is offered, how many ways exist for the department to sign up for the courses?

c. Because of availability of course leaders, it is necessary to assure that certain enrollments are met for specific days the course is offered. Administrators determined that at least 5 faculty members must take the Monday course, at least 4 faculty members must take the course on Tuesday and at least 4 on Wednesday, and there must be more than 5 enrolled for the Thursday version. How many ways now exist for the department to sign up for the courses?



Members of a 7th-grade class recently identified their interests in collecting things. Among those enrolled in the class, 15 collect stamps, 14 collect coins, and 18 collect hobby cards. Exactly 8 of the students collect both stamps and coins, 7 of them collect both coins and hobby cards, and 9 of them collect stamps and hobby cards. In the class, there are 6 students who collect stamps, coins, and hobby cards.

If at least one student enrolled in this 7th-grade class does not collect any of the items listed above, what is the smallest number of students enrolled in this class?



In the expansion of (a + b + e + d + e)^21, determine the number of different ways a coefficient of 21 appears among the collected terms.



How many ways can the letters in the word adequateness be arranged so that three consonants start the arrangement and three consonants end the arrangement?



My friend Michael Link posed this problem when we met as part of an NCTM committee. It seems that while at school one day, he met a locksmith who was trying to fix a door lock. The lock was a type you may have seen before, where the user presses buttons in a certain sequence to open the lock.

This particular lock had four pads arranged vertically, each labeled with a different letter A, B, C, and D. A valid combination for the lock met the following restrictions:

  • The combination must have exactly two inputs.
  • An input means to press from one, two, or three letters simultaneously.
  • No letter could be pressed more than once.
  • Input order is important. For example, to press "A" and then the pair "B/D" represents a different combination than to press "B/D" and then "A." However, the single input "B/D" is the same as the single input "D/B" because the two letters are pressed simultaneously.

How many different combinations exist for the lock described here?




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