Final Examination: Small-Group Portion

Combinatorics Topics For K-8 Teachers
MAT 305 Summer 1997

Roger Day


Juanita and Carlos are in a line that contains n different people, including Juanita and Carlos. If there are exactly k people between Juanita and Carlos, how many different line-ups could there be?

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A wall-mounted keypad garage door opener looks like the drawing to the right. It has four rows and three columns of keys. A 4 digit code is entered, one key push at a time, to open the garage door. Jack Sprat has been trying to determine the code for the garage door opener by watching the homeowners use it.

From inside his car parked on the street, Jack can tell when the owner moves to a new row to push a key, but he can't tell which number is pushed in any one row. Jack's observations have led him to conclude that the code is entered as follows:

  • a key in some row is pushed
  • two keys are pushed in the row immediately above the row where the first key was pushed
  • the last key is pushed in the same row as where the first key was pushed

Determine the number of different 4-digit codes Jack would need to try to be assured of entering the correct code.

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Consider the summation .

(a) Determine S1, S2, and S3.

(b) For what values of n is Sn a perfect square? Explain and justify your response.

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