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An exit poll of 320 voters yielded the following information regarding voting patterns on Propositions A, B, C, and D: 119 voters voted yes on A; 163 voted yes on B; 129 voted yes on C; 142 voted yes on D; 37 voted yes on all four; 25 voted yes on A only; 60 voted yes on B only; 59 voted yes on A and B; 70 voted yes on A and C; 82 voted yes on B and D; 93 voted yes on C and D; 10 voted yes on A, B, and C and no on D; 2 voted yes on A, B, and D and no on C; 16 voted yes on A, C, and D and no on B; and 30 voted yes on B, C, and D and no on A. How many of the surveyed voters voted no on all four propositions?

Question

An exit poll of 320 voters yielded the following information regarding voting patterns on Propositions A, B, C, and D: 119 voters voted yes on A; 163 voted yes on B; 129 voted yes on C; 142 voted yes on D; 37 voted yes on all four; 25 voted yes on A only; 60 voted yes on B only; 59 voted yes on A and B; 70 voted yes on A and C; 82 voted yes on B and D; 93 voted yes on C and D; 10 voted yes on A, B, and C and no on D; 2 voted yes on A, B, and D and no on C; 16 voted yes on A, C, and D and no on B; and 30 voted yes on B, C, and D and no on A. How many of the surveyed voters voted no on all four propositions?

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Solution

To find out how many voters voted no on all four propositions, we first need to find out how many voters voted yes on at least one proposition.

From the given data, we have:

  • 119 voters voted yes on A
  • 163 voters voted yes on B
  • 129 voters voted yes on C
  • 142 voters voted yes on D

However, these numbers include voters who voted yes on multiple propositions. To avoid double-counting, we need to subtract the number of voters who voted yes on multiple propositions:

  • 37 voters voted yes on all four propositions
  • 59 voters voted yes on A and B
  • 70 voters voted yes on A and C
  • 82 voters voted yes on B and D
  • 93 voters voted yes on C and D
  • 10 voters voted yes on A, B, and C and no on D
  • 2 voters voted yes on A, B, and D and no on C
  • 16 voters voted yes on A, C, and D and no on B
  • 30 voters voted yes on B, C, and D and no on A

Adding these numbers together, we get:

119 + 163 + 129 + 142 - 37 - 59 - 70 - 82 - 93 - 10 - 2 - 16 - 30 = 154

So, 154 voters voted yes on at least one proposition.

Since there were 320 voters in total, the number of voters who voted no on all four propositions is:

320 - 154 = 166

So, 166 of the surveyed voters voted no on all four propositions.

This problem has been solved

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