There are 10 students in a class, in how many ways can we group them in to 5 groups of 2?

To solve the problem of grouping 10 students into 5 groups of 2, we use the concept of combinations and division by repeated groupings.

### Step-by-Step Solution:

1. **Total Number of Ways to Pair Up:**
   First, calculate the number of ways to choose 2 students out of 10 for the first group. This is done using combinations:
   \[
   \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45
   \]

2. **For the Next Groups:**
   After selecting the first pair, 8 students remain. Now, we choose 2 out of these 8 for the second group:
   \[
   \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28
   \]
   Repeating this process for the remaining students:
   \[
   \binom{6}{2} = 15, \quad \binom{4}{2} = 6, \quad \binom{2}{2} = 1
   \]

3. **Multiply All the Combinations:**
   To find the total number of ways to form all the groups:
   \[
   45 \times 28 \times 15 \times 6 \times 1 = 113400
   \]

4. **Account for Group Indistinguishability:**
   Since the order of the groups doesn’t matter, we must divide by the number of ways to arrange the 5 groups. There are \(5!\) ways to arrange 5 groups, and \(5! = 120\).
   Thus, the number of ways to group the students is:
   \[
   \frac{113400}{120} = 945
   \]

### Final Answer:
There are **945 ways** to group 10 students into 5 groups of 2.

Would you like more details on this approach or have any questions?

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Here are 5 related questions:
1. How would the calculation change if the groups were distinguishable?
2. What if we wanted to form unequal-sized groups?
3. How does the number of groups affect the outcome in similar problems?
4. Can you apply this logic to partition 12 students into 4 groups of 3?
5. How does this problem relate to partition theory in combinatorics?

**Tip:** Always remember to adjust for overcounting when dealing with indistinguishable groups.

There are 10 students in a class, in how many ways can we group them into 5 groups of 2?

Explore how to group 10 students into 5 groups of 2 using combinatorics. This problem uses combination formulas and factorials to arrive at the solution, accounting for indistinguishable groups. A step-by-step solution shows that there are 945 possible ways to form the groups. Ideal for students in Grades 11-12.

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