Answer to Question #298241 in Discrete Mathematics for Danu

Question

250 members of a certain society have voted to elect a new chairman. Each member may

vote for either one or two candidates. The candidate elected is the one who polls most votes.

Three candidates x, y z stood for election and when the votes were counted, it was found that:

- 59 voted for y only, 37 voted for z only

- 12 voted for x and y, 14 voted for x and z

- 147 voted for either x or y or both x and y but not for z

- 102 voted for y or z or both but not for x

Required

i. Present the information in a Venn diagram. (6 Marks)

ii. How many voters did not vote? (4 Marks)

iii. How many voters voted for x only? (2 Marks)

iv. Who won the elections? (2 Marks)

Answer 1

x, y, z - vote for x, y, and z

xy, xz, yz - vote both x and y, x and z, y and z

n - not vote

Total members 250:

 x + y + z - xy - xz - yz + n = 250   (1)

59 voted for y only, 37 voted for z only

 y - xy - yz = 59                     (2)

 z - xz - yz = 37                     (3)

12 voted for x and y, 14 voted for x and z

 xy = 12                              (4)

 xz = 14                              (5)

147 voted for either x or y or both x and y but not for z

 x + y - xy - xz - yz = 147           (6)

102 voted for y or z or both but not for x

 y + z - xy - xz - yz = 102           (7)

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Add (2) and (3):

 y + z - xy - xz -2*yz = 96

Subtract (7):

 -yz = -6

so yz = 6

From (2):

 y = 59 + 12 + 6 = 77

From (3):

 z = 37 + 14 + 6 = 57

From (6):

 x = 147 - 77 + 12 + 14 + 6 = 102

From (1):

 n = 250 - 77 - 57 - 102 + 12 + 14 + 6 = 46

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Answers:

i

ii 46

iii x - xy - xz = 102 - 12 - 14 = 76

iv x won