1. 9/12/2015MATH 106, Section 11 MATH 106 Combinatorics “In how many ways is it possible to …?”

2. 9/12/2015MATH 106, Section 12 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?

3. 9/12/2015MATH 106, Section 13 When faced with a problem we’ve never 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 …

4. 9/12/2015MATH 106, Section 14 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?

5. 9/12/2015MATH 106, Section 15 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

6. 9/12/2015MATH 106, Section 16 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 …

7. 9/12/2015MATH 106, Section 17 Try it for 5 steps Change the steps into a rectangle: 3 by 5 That 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 + 1 This is . 2

8. 9/12/2015MATH 106, Section 18 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. n This is . 2 This is n + 1.

9. 9/12/2015MATH 106, Section 19 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 oddn is evenn = 25n = 100 n+1 ——  n 2 any n n —  (n+1) 2 n(n+1) ——— 2 3255050

10. 9/12/2015MATH 106, Section 110 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. RRRBRBBRBRBB4 arrangements RRRRRBRBRRBRRBB 8 arrangements BRRBRBBRBBBRBBB RRRRRRRBRRBRRRBBRBRRRBRBRBRBRBBRRBBB BRRRBRRBBRBRBRBRBRBB 16 arrangements BBRRBBRBBBBRBBBB 2 n arrangements #1

11. 9/12/2015MATH 106, Section 111 How many different color arrangements are possible with eight steps? 2 8 = 256 arrangements

12. 9/12/2015MATH 106, Section 112 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. RBRBBRBR2 arrangements PRBPRBRPBRPBRBPRBPPBRPBR6 arrangementsBPRBPRBRPBRP GPRBGPRBPGRBPGRBPRGBPRGBPRBGPRBGGRPBGRPBRGPBRGPBRPGBRPGBRPBGRPBG 24 arrangements n(n–1)(n–2)…(3)(2)(1) arrangements …etc. #2

13. 9/12/2015MATH 106, Section 113 How many different arrangements are possible with eight different colored flags? (8)(7)(6)(5)(4)(3)(2)(1) = 40320 arrangements

14. 9/12/2015MATH 106, Section 114 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.