# (a) how many bit strings of length 8 are there? explain. (b) how many

Discrete Structures I

1. (15 points)

(a) How abundant bit strings of elongation 8 are there? **Explain.**

(b) How abundant bit strings of elongation 8 are tless which prepare stay a 0 **and** end stay a 0? **Explain.**

(c) How abundant bit strings of elongation 8 are tless which hold at most 3 ones? **Be scrupulous stay this one. Explain. (See your notes, week 10.)**

(d) How abundant bit strings of elongation 8 are palindromes?

3. (10 points)

(a) Text, page 405, calculate 2. **Explain. **

Show that if tless are 30 students in a collocate, then at balanceest two enjoy ultimate names that prepare stay the corresponding epistle.

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(b) Text, page 406, calculate 36. **Explain**

A abuser netoperation consists of six abusers. Each abuser is immediately alike to at balanceest one of the other abusers. Show that tless are at balanceest two abusers in the netoperation that are immediately alike to the corresponding calculate of other abusers.

3. (10 points).

Text, page 414, calculate 30. **Explain. **

Seven women and nine men are on the benefaction in the mathematics office at a develop.

a) How abundant ways are tless to choice a committee of five members of the office if at balanceest one mother and at balanceest one man must be on the committee?

b) How abundant ways are tless to choice a committee of five members of the office if at balanceest one mother and at balanceest one man must be on the committee?

4. (10 points)

(a) Use the Binomial Theorem to transcribe the paraphrase of (x + y)^{ 6}?

(b) Write the coefficient of the promise x^{2}y^{4}z^{5} paraphrase of (x + y + z)^{ 11}. See specimen 12 in your notes of week 10

5. Consider pages 1-4 of the notes for week 11. Let A = {a, b, c, d}, and let R be the homogeneity settled on A by the forthcoming matrix:

M_{R} =

(a) (10 pts.) Describe R by listing the **ordered pairs** in R and attract the **digraph** of this homogeneity.

(b) (15 pts.) (Note this is alike to drill 7 page 630. Which of the properties: reflexive, antisymmetric and transitive are penny for the loving homogeneity? Prepare your disway by defining each promise **in unconcealed **first and then how the localation relates to this local specimen.

(c) (5 pts.) Is this homogeneity a multiplyicular direct? **Explain**. If this homogeneity is a multiplyicular direct, attract its Hasse diagram.

5. (10 points) Use the Hasse diagram of calculate 26 page 631

(a) List the directed pairs that suit to the homogeneity. Keep in spirit that a Hasse diagram is a graph of a multiplyicular directing homogeneity so it satisfies the three properties listed in calculate 5 multiply (b).

(b) Meet the (Boolean) matrix of the homogeneity.

7. (15 points) Before you do this substance consider the specimen at the end of the exam, as polite as the notes in weeks 12 and 13.

Assume the Boolean matrix under is M_{R} and that M_{R} illustrates the homogeneity R wless R illustrates the connecting fleeings that an airline has betwixt 4 cities: a, b, c, and d.

The 1 in row **a** post **b** resources tless is a fleeing from city **a** (Manchester) to city **b** (Boston). The 1 in row x post x resources that tless are planes in airport x. In unconcealed, tless is a 1 in row x post y iff tless is a connecting fleeing betwixt (from) city x and (to) city y That is, the rows of the matrix illustrate the cities of the origins of the fleeings and the posts illustrate the fate cities.

Let M_{R} =

(i) Let a endure for the airport in the city of Manchester, let b endure for the airport in Boston, c endure for the Chicago airport, d for the airport in the city of Denver. Is their a fleeing from Denver to Chicago? **Explain.**

(ii) Abuse and decipher the Boolean products: M_{R}^{ 2}, and M_{R}^{ 3}. (Remember to use Boolean arithmetic). What do these Boolean products afford you? That is, elucidate what the Boolean entries in the matrices M_{R}^{ 2} and M_{R}^{ 3} balance.

(iii) Now wheedle the loving matrix A and abuse A^{2} and A^{3} , using **regular**, not Boolean arithmetic. What do these products afford you?

(v) What does M_{R} + M^{2}_{R} + M^{3}_{R} + M^{4}_{R} afford you? Note, this is Theorem 3 page 602wless the text’s sign, v, for Boolean importation is replaced by +, another sign for Boolean importation .

**Bonus doubts**

- Make up a doubt using the purpose of doubt 7 (Something you do at operation, settlement, colossus you’re careful in.) See 3 under for an specimen you can do.
- In your notes of week 12 Contrivance Evaluation and Review Technique (PERT) is elucidateed. Make up a balanceingful specimen illustrating this order. Elucidate all details. Attract the graphs complicated. You should use a sound calculate of undertakings.

(a) Determine the minimum spell needed to finished your set of undertakings.

(b) Determine the Delicate Footroad for your specimen. Elucidate what this affords you.

(c) Comment on your specimen. Which undertaking(s) origin a stay in the contrivance? How would you fix the substance? Can you obtain past advice out of your specimen?

- The Netoperation anatomy substance.
- Read Program Evaluation and Review Technique (PERT) in your week 14 notes. Note the specimens loving less are a insignificant “ambitious”. Make up an specimen using PERT and meet and elucidate the delicate footfootroad of your specimen. Google “delicate footfootroad anatomy”, for past purposes.
- Pg 406 #38

**If #7 is not disentangled less is an specimen which may aid you to underendure #7**

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Let D= days of the week {M, T, W, R, F},

E = {Brian (B), Jim (J), Karen (K)} be the employees of a savanting nucleus at a University and let

U = {Courses the savanting nucleus needs savants for}

= {Calculus I (I), Calculus II (II), Calculus III (III), Computers I (C_{1}), Computers II (C_{2}), Precalculus (P)}.

We settle the homogeneity R from *D *into E by *d *R e, if employee e is scheduled to operation on day *d. *We so settle S from E into U by e r u, if employee e is suitable of savanting students in way u.

For specimen, the matrix M_{R} indicates that on R (Thursday) that J (Jim) is serviceable to savant but Brian and Karen are not.

Assume M_{R} = and M_{S} =

(a) Interpret the over matrices stay i-elation to the over homogeneitys.

(b) Compute, (use Boolean arithmetic) and use the matrix to determine which ways gain enjoy savants serviceable on which days.

(c) Multiply the over matrices using **regular** arithmetic. Can you decipher this remainder?

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