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Mathematical
Concepts
Discussion Topic 3
Counting and Pascal’s Triangle
General Instructions
 Meet and discuss the topic with the members in your group. You may do some
research ahead of time on individual topics, but remember it is not a major component
of this part of the course.
 There are no right answers to these problems and every student has something
valuable to contribute based on their own ideas and experiences.
 The group will have one week to submit a one-page (no more) report highlighting their
discussion. Include a list of interesting references (if you find any) on the topics (like
websites or articles). WARNING: A group’s written summary will not be accepted after
the class discussion has taken place.
 There will also be a class discussion where each group contributes some of their ideas
on the topic.
 The goal of the group work is to help develop mathematical thinking and
communication.
 Your grade will be based on the written report, peer evaluation and group and individual
class contributions.
 Everyone in the group should sign the summary before it is submitted and groups
should sit together during class discussions on the topic.
Topic: Counting and Pascal’s Triangle
For this discussion you will need the version of Pascal’s Triangle you created in September.
As a group, you will be looking at several counting problems. Sometimes you will be asked to
find general methods to solve them, sometimes you will be asked to find their connection with
Pascal’s Triangle, sometimes both.
1) a. How many ways are there to arrange all four letters of MATH? List them making sure
not to list any more that once. Guess at a general formula for arranging four letters.
b. Using your formula how many ways are there to arrange all of PETE? List them making
sure not to list any more that once. If you were wrong modify your guess at a general
formula for arranging four letters.
c. How ways does your formula are there to arrange all of EEEK? NOON? List them making
sure not to list any more that once. If you were wrong modify your guess at a general
formula for arranging four letters.
d. Make a guess at a general formula for arranging n letters where some of the letters may
be the same. Use AAAHH, to check your answer. How many ways can arrange the
letters ARRANGE?
2) Work out the numerical value of C(m, l) for several small values of m and l. So for example,
find C(1,0)= ,C(1,1)= , C(2,0)= , C(2,1)= , C(2,2)= , C(3,0)= ,… Use these to try to
establish how these patterns fits into Pascal’s Triangle.
3) Find and simplify (x+1)0, (x+1), (x+1)2 and (x+1)3. (Ask how if no one in your group knows
how to find one or all of these). The coefficients of these equations are the numbers in
front of the numbers in front of variable terms. For example the coefficient of 3x 2 is 3. How
do the coefficients of these four equations relate to the first four rows of Pascal’s triangle?
What would (x+1)4 be?
4) The holiday season is coming up and you will be away at Christmas time so you decide to
send your mother some roses to brighten her holiday. You decide you want to send her
some number of roses (n roses) and the flower shop you order from has d different colours
of roses to choose from (where d is some number). You will phone the order into the shop
and they make the arrangement for you so the order you select the roses doesn’t matter.
Your job is to find a general formula (in terms of n and d) for the number of different
possible ways you can request the roses. Use the following steps to help you:
a. Work out the number of ways to do this task for several small values of r and d. It may
be helpful to start with d=1 (that is the flower shop only has red roses) and work out
the problem for n=1, and then work out the problem for n=2, etc. Then move on to try
d=2 (that is the flower shop has only red and yellow roses) and work out the problem
for n=1, and then work out the problem for n=2, etc. You should also see what
happens when d=3 and n=1, n=2 or n=3.
b. Try to see how these numbers fit in with Pascal’s Triangle. You may need to find
some more values if you are having trouble here.
c. Use your answers to question 2 and 4 b. to establish a formula of the form C(???,???)
where the question marks are filled in with formulas in terms of n and d.
These questions are meant to be done as a group and are of a higher difficulty level. Get
started on them as soon as possible. Make sure you double-check each other’s work and that
you understand the questions being asked. A small mistake can lead you astray. Feel free to
come by during office hours and ask questions. Your report should discuss the answers to
the problems and how they were reached. Not every detail needs to be included. Focus your
report on the pattern recognition and the final results of each problem.