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Let's analyze the problem step by step.

### Part A: Expression for the Function
The problem shows a table where the number of people at a meeting (x) corresponds to the number of handshakes (y). The goal is to find the analytical expression that represents this function.

We are given:
- \( x = 2, y = 1 \)
- \( x = 3, y = 3 \)
- \( x = 4, y = 6 \)
- \( x = 5, y = 10 \)
- \( x = 6, y = 15 \)
- \( x = 7, y = 21 \)
- \( x = 8, y = 28 \)
- \( x = X, y = 190 \)

Looking at the values, the number of handshakes seems to follow a quadratic pattern. In fact, the number of handshakes between \( x \) people can be represented by the formula:

\[
y = \frac{x(x-1)}{2}
\]

This formula comes from the combination formula for choosing 2 people out of \( x \) to shake hands, i.e., \( C(x, 2) \).

### Part B: Handshakes for 12 People
Now, we can use the formula \( y = \frac{x(x-1)}{2} \) to calculate the number of handshakes when \( x = 12 \):

\[
y = \frac{12(12-1)}{2} = \frac{12 \times 11}{2} = \frac{132}{2} = 66
\]

So, in a meeting of 12 people, there are 66 handshakes.

### Part C: Find \( x \) for 190 Handshakes
We are asked to find the number of people \( x \) when there are 190 handshakes. Using the formula:

\[
\frac{x(x-1)}{2} = 190
\]

Multiplying both sides by 2 to eliminate the fraction:

\[
x(x-1) = 380
\]

Solving the quadratic equation:

\[
x^2 - x - 380 = 0
\]

Now, we can solve this using the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-380)}}{2(1)}
\]
\[
x = \frac{1 \pm \sqrt{1 + 1520}}{2}
\]
\[
x = \frac{1 \pm \sqrt{1521}}{2}
\]
\[
x = \frac{1 \pm 39}{2}
\]

So, \( x = \frac{1 + 39}{2} = 20 \) (since the negative solution doesn't make sense).

Thus, the number of people is \( x = 20 \) when there are 190 handshakes.

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Do you have any questions, or would you like further details on any part of the solution?

### Related Questions:
1. What is the general formula for the number of handshakes if there are \( x \) people?
2. How does the quadratic equation relate to this type of handshake problem?
3. How can the combination formula \( C(x, 2) \) be applied in other real-world scenarios?
4. How would the number of handshakes change if the rule was to shake hands only with non-immediate neighbors?
5. What are some other scenarios where quadratic relationships appear?

#### Tip:
When solving a quadratic equation, always consider whether both the positive and negative solutions make sense in the context of the problem.

In a meeting, all attendees shake hands with each other. The number of handshakes depends on the number of attendees, according to the table. Part A: Find the analytical expression for the function. Part B: Calculate the number of handshakes for 12 attendees. Part C: Find the number of attendees when there are 190 handshakes.

Learn how to solve a handshake problem where all attendees shake hands, using combinatorics and quadratic equations. This includes finding an analytical expression for handshakes, calculating for 12 attendees, and solving for the number of people when there are 190 handshakes. Suitable for Grades 9-11 students.

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