Classification of Combinatorial problems

1. Classification of Combinatorial problems i) Order matters / does not matter • Choose a committee of 3 out of 10 members • (members are distinguishable): the order of the • members in committee does not matter C (10, 3) • Choose president, vice-president and secretary • out of 10 members: three different assignments mean • the order is important 1098

2. ii) identical objects • Count different arrangements of 4 objects: AABC Modify the problem: AaBC Then we have 4!=24 distinguishable arrangements For any positions of B and C we have 2 choices, that should not be distinguished. So only 24/2=12 arrangements are distinct • Count arrangements of AABB 12/2=6

3. How many distinct 3-letter passwords exist? (order is important abcbac) Any letter can be used only once (without repetition) many times (with repetition) iii) with/without repetitions 262524 263 • You draw 3 balls from a box with 10 red, 10 blue and 10green (order is not important: rbg=brg=gbr=… ) *|*|* **||* ||***

4. A committee of 3 is to be chosen from 5 Democrats, 3 Republicans and 4 Independents. a) In how many ways it can be done? • all representatives are not distinguished • all members of a committee are equivalent • selections with repetition • the same problem: 5 red, 3 blue and 4 green balls are • picked at random. How many different outcomes are possible? **|*| DDD IDD DDR IDR DRR IRR RRR IIR I I I IID

5. **||* *|*|* *||** |**|* |*|** ||*** | |* D I R Consider the situation, when all representatives are distinguished. What will be the number of possible committees? C (12, 3) b) In how many ways it can be done if the committee must contain at least one independent? We must put one star in the “independent box” and count different ways to distribute two remaining stars in three boxes. DDI DRI DII RRI RII III

6. For what situations the following answers apply ? 3 distinguished positions in the committee and all representatives are distinguished 121110 12 distinguished positions in the committee, representatives of a party are not distinguished

7. In how many ways can we select three books each from different subject from a set of 6 distinct history books, 9 distinct classic books, 7 distinct law books, and 4 distinct education books? C (6,3)  C (9,3)  C (7,3)  C (4,3) An exam has 12 problems. How many ways can integer points be assigned to the problems if the total of the points is 100 and each problem is worth at least five points? x1+ x2 +…+ x12=100 ( xi 5 )

8. q r q # r 1 1 1 1 0 1 0 1 0 0 0 1 Find the formula involving the connectives , , and  that has the following truth table: You can observe that q # r  (rq)  rq

9. Let ? be an unknown boolean logical operator. The logical statement [(p q) r]  (q ? r) is equivalent to [(pq) r]. Given this information, there are 2 possible truth tables for the boolean logical operator ?. List both of these truth tables. q ? r 1 0 1 1/0 1 1/0 1 1/0 p q r pq (p q)r 1 11 1 1 1 1 0 1 1 1 0 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 1 0 1 00 0 0 0 (pq) r [(p q) r]  (q ? r) 1 0 1 1 1 1 1 1

10. q r q ? r 1 1 1 1 0 0 0 1 1 0 0 0/1 Answer:

11. Use laws of logic to show that the following expression is a tautology: p  [( pq )  (q  r )] ( q  r )  (q p)  (q  r ) ( pq )  (q  r )  q  (p  r ) [( pq )  (q  r )] ( q  r )  [q  (p  r )]  ( q  r )  (q  q) (p  r )  r  T (p  r )  r  T p  T T

12. x [ p(x) q (x) ] [x p(x)] [x q (x) ] x [ p(x)  q (x) ]  x [ p(x)  q (x) ] [x p(x)][x q (x) ] [ x p(x)][ x q (x) ] x [ p(x) q (x) ] [x p(x)]  [x q (x) ] In each case determine whether or not two propositions are logically equivalent:    

13. C C B A A B C A Goal Given C B  C A A  B Suppose A  B and C is any set.Prove or disprove that C B  C A. Proof.

14. A  B x  C A Goal Given C B  C A A  B x  C B  x  C A Contrapositive: x  C A  x  C B x  C B   (x C  xA)  (x  C  x  A)  (x  C  x  B)  (x  C  x  B)  x  C B