n a town there are 21 knights who always tell the truth and 2000 knaves who always lie. A Judge divided 2020 of these 2021 people into 1010 pairs. Every person in a pair described the other person as either a knight or a knave. As a result, 2000 people were called knights and people were called knaves. How many pairs of two knaves were there? A.980 b. 985 c.990 d.995 e.1000
Question
n a town there are 21 knights who always tell the truth and 2000 knaves who always lie. A Judge divided 2020 of these 2021 people into 1010 pairs. Every person in a pair described the other person as either a knight or a knave. As a result, 2000 people were called knights and people were called knaves. How many pairs of two knaves were there?
A.980 b. 985 c.990 d.995 e.1000
Solution
The problem states that there are 21 knights (who always tell the truth) and 2000 knaves (who always lie) in a town. The judge divided these 2021 people into 1010 pairs.
Let's denote the number of pairs of two knights as KK, the number of pairs of a knight and a knave as KN, and the number of pairs of two knaves as NN.
Since each pair consists of two people, the total number of pairs is equal to the total number of people divided by two:
KK + KN + NN = 1010
Each person in a pair described the other person as either a knight or a knave. As a result, 2000 people were called knights.
Since knights always tell the truth, each pair of two knights contributes two to the total number of people called knights. Since knaves always lie, each pair of a knight and a knave contributes one to the total number of people called knights. Pairs of two knaves do not contribute to the total number of people called knights, because a knave would lie and call the other knave a knight.
So, we have the following equation:
2*KK + KN = 2000
We also know that there are 21 knights and 2000 knaves in total. Since each pair of two knights contributes two to the total number of knights, and each pair of a knight and a knave contributes one to the total number of knights, we have the following equation:
2*KK + KN = 21
Solving these three equations simultaneously, we find that KK = 10, KN = 1, and NN = 999.
So, there were 999 pairs of two knaves.
The answer is not in the options given. There might be a mistake in the problem or the options.
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