04/19/23 MATH 106, Section 1 1

MATH 106

Combinatorics

“In how many ways is it possible to …?”

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Such as …

• How many solutions in positive integers are there to the equation a + b + c + d = 52?

• In how many ways can you make change for a dollar?

• You are taking your friend home to meet your parents. How many possible seating arrangements are there around the dinner table?

• How many seating arrangements are there if you and your friend are seated across from each other?

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When faced with a problem we’venever seen before …

• We can try to answer the problem directly with some formula, or “recipe.”

• If that doesn’t work, we can ask ourselves if the problem is similar to one we already know how to do?

• If these attempts fail, try to solve an easier version of the problem. If successful, try to work up to the original problem. Look for what’s known as a “general solution” – that is, one that will work in all cases of this problem.

• Let’s try this third method …

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Building a narrow staircase

Problem: A narrow staircase, one foot wide, is to be built out of concrete blocks. Each block is a one foot cube, and the space underneath the steps is to be filled in as a massive wall of concrete blocks. How many blocks are necessary to construct a staircase with ten steps?

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Let’s start with a small case

How many would it take for 1 step? 2 steps? 3 steps? and so on?

For 10 steps …1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55

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Can we “generalize” that result?

• How many blocks would it take for n steps?• 1 + 2 + 3 + … + (n – 1) + n

• That’s the right idea, but what if we have 100 steps? That’s a lot of addition. Can we do better?

• Some of you may know a formula for getting the sum of integers from 1 to n

• Let’s see if we can derive it from this problem …

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Try it for 5 steps

Change the steps into a rectangle: 3 by 5That gives us 15 again. That’s good!

We’ve computed the answer by a different method and got the same answer. This is an example of a combinatorial proof.

This is n.

n + 1This is . 2

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Try it for 6 steps

Change the steps into a rectangle: 3 by 7

Work in groups to come up with a general rule for n steps. Hint: Do the cases where n is odd and n is even, separately.

nThis is . 2

This is n + 1.

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Let’s try this method on other problems …

So what formula did you come up with. How many blocks are needed for 25 steps? 100 steps?

n is odd n is even n = 25 n = 100 n+1—— n 2

any n n— (n+1) 2

n(n+1)——— 2

325 5050

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Each step in a spiral staircase is to be painted with one of the two colors red (R) or blue (B). How many different color arrangements are possible with two steps? three steps? four steps?

Come up with a general rule for n steps.

RR RB BR BB 4 arrangements

RRR RRB RBR RBB8 arrangements

BRR BRB BBR BBB

RRRR RRRB RRBR RRBB RBRR RBRB RBBR RBBB

BRRR BRRB BRBR BRBB16 arrangements

BBRR BBRB BBBR BBBB

2n arrangements

#1

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How many different color arrangements are possible with eight steps?

28 = 256 arrangements

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Different colored flags are to be flown on a flagpole with one color on top, a different color underneath, down to the last different color on the bottom. How many different arrangements are possible with two different colored flags? three different colored flags? four different colored flags?

Come up with a general rule for n different colored flags.

RB BR 2 arrangements

PRB RPB RBP PBR 6 arrangementsBPR BRP

GPRB PGRB PRGB PRBG GRPB RGPB RPGB RPBG

24 arrangements

n(n–1)(n–2)…(3)(2)(1) arrangements

…etc.

#2

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How many different arrangements are possible with eight different colored flags?

(8)(7)(6)(5)(4)(3)(2)(1) = 40320 arrangements

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Homework Hints:

In Section 1 Homework Problem #2,

In Section 1 Homework Problem #4,

construct a two-column table, where the first column is the number of people attending and the second column is the number of handshakes.

note the similarity with #2 on the Section#1 class handout.