Mr. Trump invites three married couples to his penthouse for dinner. Upon arriva

Question

Mr. Trump invites three married couples to his penthouse for dinner. Upon arrival, Mr. Trump and the six guests shake hands of some of the other people (none of the guests shakes hands with their spouse). Suppose each of the six guest shakes a different number of hands (possibly one person shakes no hands). In graph terms, we have seven vertices with six of them, the guest vertices, paired off into three pairs and no edge between two paired vertices. Each of the six paired vertices has a different degree. What is the set of the different degrees of the six guest vertices Explain. Did Mr. Trump shake an even or an odd number of hands (is Trump's vertex of even or odd degree)? Explain. Exactly how many hands did Mr. Trump shake? Explain

Explanation / Answer

a)Total no people are 7 i.e 6 guests + Trump

Max shake hands a guest can give is 5 i.e (7- himself-his/her patner )

Min no of shake hands a guest can give is 0

there are 6 different numbers between between 0 and 5 (inclsusive)

so set of degress of six guests can only these 6 numbers between 0 and 5 ie {0,1,2,3,4,5}

b)

sum of degress of all guests is (1+2+3+4+5) = 15

let the degree of trump be T

sum of all degress of all people is 2*no of hand shakes beause each shake hand counts to two vetcices (one for person 1 and other for person 2 )

since sum of degres of all vertices is even

15+T is even

so T is odd

degree of trump of trump vertex is odd

c)sum of degress among guests not involving trump = 15-t

no of sahke hands among guests = (15-t)/2

let the represent guess by G0,G1,G2,G3,G4,G5

Gi -gets i shake hands

G5 must give shake hand to trump to make his count as 5since he can only shake hands to G1,G2,G3,G4 among guests

fact 1: so G0 is patner of G5 since he no one from G1,G2,G3,G4 is his/her patner as he is giving shake hand to everyone.

fact 2: G1 already got one shake hand so he cannot get anymore shake hands

G4- can give shake hand to only 3 among (G5,G2,G3).so trump will get one shake hand from G4

fact 3: so G4 is patner of G1

fact 4: G2 already got two shake hands from G5 and G4 .so he/she cannot get anymore shake hands.

G3-who already got two sahe hands from G4 and G5 can only give his reamining shake hand to trump.

so trump will get three shake hands (from G5,G4,G3).

Related Questions

Navigate

• Browse (All)
• Browse M
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.