Three different numbers from 1 to 20 are chosen. They add to 20 Two of the numbers are 2 apart. How many possibilities are there for the largest number?

To find how many possibilities there are for the largest number among three different numbers from 1 to 20 that add to 20, with two of the numbers being 2 apart, we can proceed as follows:

Let the three numbers be \(a\), \(b\), and \(c\) such that \(a < b < c\), and we are given:
1. \(a + b + c = 20\)
2. \(b = a + 2\) (since two of the numbers are 2 apart)

Substitute \(b\) in the first equation:

\[
a + (a + 2) + c = 20
\]

Simplify this to:

\[
2a + 2 + c = 20
\]

Subtract 2 from both sides:

\[
2a + c = 18
\]

Rearrange to solve for \(c\):

\[
c = 18 - 2a
\]

We must ensure \(a\), \(a+2\), and \(18-2a\) are all distinct numbers and within the range of 1 to 20. Also, since \(a < b < c\):

\[
a < a + 2 < 18 - 2a
\]

Let's solve the inequality \(a + 2 < 18 - 2a\):

\[
a + 2 < 18 - 2a
\]

Add \(2a\) to both sides:

\[
3a + 2 < 18
\]

Subtract 2 from both sides:

\[
3a < 16
\]

Divide by 3:

\[
a < \frac{16}{3}
\]

Since \(a\) must be an integer, the largest possible value for \(a\) is 5. So \(a\) ranges from 1 to 5.

Next, we calculate \(b\) and \(c\) for each possible \(a\):

1. For \(a = 1\):
   \[
   b = a + 2 = 3
   \]
   \[
   c = 18 - 2a = 18 - 2 \times 1 = 16
   \]

2. For \(a = 2\):
   \[
   b = a + 2 = 4
   \]
   \[
   c = 18 - 2a = 18 - 2 \times 2 = 14
   \]

3. For \(a = 3\):
   \[
   b = a + 2 = 5
   \]
   \[
   c = 18 - 2a = 18 - 2 \times 3 = 12
   \]

4. For \(a = 4\):
   \[
   b = a + 2 = 6
   \]
   \[
   c = 18 - 2a = 18 - 2 \times 4 = 10
   \]

5. For \(a = 5\):
   \[
   b = a + 2 = 7
   \]
   \[
   c = 18 - 2a = 18 - 2 \times 5 = 8
   \]

Thus, the corresponding values for \(c\) (the largest number) are 16, 14, 12, 10, and 8.

Therefore, there are 5 possibilities for the largest number.

Would you like more details or have any questions about this solution? Here are some related questions you might find interesting:
1. What if the range was from 1 to 30?
2. How many possibilities are there if the sum was different?
3. How would the answer change if the difference was more than 2?
4. What if we were looking for combinations where one number is double the other?
5. Can we generalize this problem for different sums and differences?

Tip: Always check for the constraints and conditions given in a problem to simplify your solution approach.

Discover how to find the number of possibilities for the largest number among three different numbers from 1 to 20 that sum to 20, with two of the numbers being 2 apart. Explore the detailed step-by-step solution using concepts from number theory and linear equations. Suitable for students in Grades 7-9.

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