In A Certain Game, Each Token Has One Of Three Possible Values: 1 Point, 5 Points, Or 10 Points. How Many Different Combinations Of These Token Values Are Worth A Total Of 17 Points? Justify Your Choice.A. Two B. Three C. Four D. Five E. Six

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Introduction

In a game where tokens have different point values, players can combine these tokens to achieve a specific total score. In this scenario, we are tasked with finding the number of different combinations of token values that are worth a total of 17 points. The tokens have three possible values: 1 point, 5 points, or 10 points. To approach this problem, we will use a combination of mathematical reasoning and logical analysis.

Understanding the Token Values

Before we dive into the combinations, let's understand the token values and their possible combinations. We have three token values: 1 point, 5 points, and 10 points. We can represent these values as follows:

  • 1 point token: 1x
  • 5 points token: 5x
  • 10 points token: 10x

where x represents the number of tokens of each value.

Analyzing Combinations

To find the number of combinations that sum up to 17 points, we can start by listing the possible combinations of tokens that add up to 17 points. We can use a systematic approach to list all possible combinations.

Combinations with 1 Point Tokens

Let's start by considering combinations that include 1 point tokens. We can have the following combinations:

  • 1x (1 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token) = 5 points (not enough)
  • 1x (1 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token) = 4 points (not enough)
  • 1x (1 point token) + 1x (1 point token) + 1x (1 point token) = 3 points (not enough)
  • 1x (1 point token) + 1x (1 point token) = 2 points (not enough)
  • 1x (1 point token) = 1 point (not enough)

We can see that we need at least 6 1 point tokens to reach 17 points, but this is not a valid combination since we are limited to 3 token values.

Combinations with 5 Point Tokens

Next, let's consider combinations that include 5 point tokens. We can have the following combinations:

  • 1x (5 point token) + 1x (5 point token) + 1x (5 point token) + 1x (5 point token) + 1x (5 point token) + 1x (5 point token) = 30 points (too much)
  • 1x (5 point token) + 1x (5 point token) + 1x (5 point token) + 1x (5 point token) + 1x (5 point token) = 25 points (too much)
  • 1x (5 point token) + 1x (5 point token) + 1x (5 point token) + 1x (5 point token) = 20 points (too much)
  • 1x (5 point token) + 1x (5 point token) + 1x (5 point token) = 15 points (too much)
  • 1x (5 point token) + 1x (5 point token) = 10 points (too much)
  • 1x (5 point token) = 5 points (not enough)

We can see that we need at least 4 5 point tokens to reach 17 points, but this is not a valid combination since we are limited to 3 token values.

Combinations with 10 Point Tokens

Finally, let's consider combinations that include 10 point tokens. We can have the following combinations:

  • 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) = 100 points (too much)
  • 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) = 90 points (too much)
  • 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) = 80 points (too much)
  • 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) = 70 points (too much)
  • 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) = 60 points (too much)
  • 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) = 50 points (too much)
  • 1x (10 point token) + 1x (10 point token) + 1x (10 point token) + 1x (10 point token) = 40 points (too much)
  • 1x (10 point token) + 1x (10 point token) + 1x (10 point token) = 30 points (too much)
  • 1x (10 point token) + 1x (10 point token) = 20 points (too much)
  • 1x (10 point token) = 10 points (not enough)

We can see that we need at least 2 10 point tokens to reach 17 points, but this is not a valid combination since we are limited to 3 token values.

Combinations with 1, 5, and 10 Point Tokens

Now, let's consider combinations that include all three token values. We can have the following combinations:

  • 1x (10 point token) + 1x (5 point token) + 1x (1 point token) + 1x (1 point token) = 17 points
  • 1x (10 point token) + 1x (5 point token) + 1x (1 point token) = 16 points (not enough)
  • 1x (10 point token) + 1x (5 point token) = 15 points (not enough)
  • 1x (10 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token) = 15 points (not enough)
  • 1x (10 point token) + 1x (1 point token) + 1x (1 point token) = 12 points (not enough)
  • 1x (10 point token) + 1x (1 point token) = 11 points (not enough)
  • 1x (5 point token) + 1x (5 point token) + 1x (5 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token) = 22 points (too much)
  • 1x (5 point token) + 1x (5 point token) + 1x (5 point token) + 1x (1 point token) + 1x (1 point token) = 17 points
  • 1x (5 point token) + 1x (5 point token) + 1x (5 point token) + 1x (1 point token) = 16 points (not enough)
  • 1x (5 point token) + 1x (5 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token) = 16 points (not enough)
  • 1x (5 point token) + 1x (5 point token) + 1x (1 point token) = 11 points (not enough)
  • 1x (5 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token) = 10 points (not enough)
  • 1x (5 point token) + 1x (1 point token) + 1x (1 point token) = 7 points (not enough)
  • 1x (5 point token) + 1x (1 point token) = 6 points (not enough)

Q: What is the problem we are trying to solve?

A: We are trying to find the number of different combinations of token values that are worth a total of 17 points. The tokens have three possible values: 1 point, 5 points, or 10 points.

Q: How did you approach this problem?

A: We used a combination of mathematical reasoning and logical analysis. We started by listing the possible combinations of tokens that add up to 17 points, and then we systematically analyzed each combination to see if it was valid.

Q: What are the possible combinations of tokens that add up to 17 points?

A: The possible combinations of tokens that add up to 17 points are:

  • 1x (10 point token) + 1x (5 point token) + 1x (1 point token) + 1x (1 point token)
  • 1x (10 point token) + 1x (5 point token) + 1x (1 point token)
  • 1x (10 point token) + 1x (5 point token)
  • 1x (10 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token)
  • 1x (10 point token) + 1x (1 point token) + 1x (1 point token)
  • 1x (10 point token) + 1x (1 point token)
  • 1x (5 point token) + 1x (5 point token) + 1x (5 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token)
  • 1x (5 point token) + 1x (5 point token) + 1x (5 point token) + 1x (1 point token)
  • 1x (5 point token) + 1x (5 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token)
  • 1x (5 point token) + 1x (5 point token) + 1x (1 point token)
  • 1x (5 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token)
  • 1x (5 point token) + 1x (1 point token) + 1x (1 point token)
  • 1x (5 point token) + 1x (1 point token)

Q: How many valid combinations are there?

A: There are 4 valid combinations:

  • 1x (10 point token) + 1x (5 point token) + 1x (1 point token) + 1x (1 point token)
  • 1x (5 point token) + 1x (5 point token) + 1x (5 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token)
  • 1x (5 point token) + 1x (5 point token) + 1x (5 point token) + 1x (1 point token)
  • 1x (5 point token) + 1x (5 point token) + 1x (1 point token) + 1x (1 point token) + 1x (1 point token)

Q: What is the total number of combinations that add up to 17 points?

A: The total number of combinations that add up to 17 points is 4.

Q: What is the final answer?

A: The final answer is 4.

Conclusion

In this article, we explored the problem of finding the number of different combinations of token values that are worth a total of 17 points. We used a combination of mathematical reasoning and logical analysis to approach this problem. We listed the possible combinations of tokens that add up to 17 points and systematically analyzed each combination to see if it was valid. We found that there are 4 valid combinations.