There are 14, 12 and 16 students in group *A*, group *B* and group *C* respectively. If 3 students are randomly selected from the 3 groups, find the probability that they come from the same group.

To find the probability that 3 students selected randomly from groups *A*, *B*, and *C* all come from the same group, we can break the problem down step by step.

### Step 1: Total number of students
- Group *A* has 14 students.
- Group *B* has 12 students.
- Group *C* has 16 students.

So, the total number of students from all 3 groups combined is:
\[
14 + 12 + 16 = 42 \text{ students in total}.
\]

### Step 2: Total number of ways to select 3 students
The total number of ways to select 3 students from 42 students can be calculated using combinations:
\[
\binom{42}{3} = \frac{42 \times 41 \times 40}{3 \times 2 \times 1} = 11480 \text{ ways}.
\]

### Step 3: Number of favorable outcomes (all students from the same group)
We need to count the number of ways to select 3 students from each group:
- From group *A* (14 students): 
\[
\binom{14}{3} = \frac{14 \times 13 \times 12}{3 \times 2 \times 1} = 364 \text{ ways}.
\]
- From group *B* (12 students): 
\[
\binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \text{ ways}.
\]
- From group *C* (16 students): 
\[
\binom{16}{3} = \frac{16 \times 15 \times 14}{3 \times 2 \times 1} = 560 \text{ ways}.
\]

Now, we add up all the favorable outcomes:
\[
364 + 220 + 560 = 1144 \text{ favorable outcomes}.
\]

### Step 4: Probability of selecting 3 students from the same group
The probability is the ratio of favorable outcomes to total outcomes:
\[
P(\text{same group}) = \frac{1144}{11480} \approx 0.0996.
\]

So, the probability that the 3 students selected come from the same group is approximately **0.0996**, or **9.96%**.

---

Would you like further details or explanations? Here are some related questions:

1. How would the probability change if there were more students in one of the groups?
2. How do you calculate the probability for students to come from different groups?
3. What happens if we select 4 students instead of 3?
4. How does the concept of combinations help in solving probability problems like this?
5. What would the probability be if there were only two groups?

**Tip:** Always check if the problem is asking for combinations or permutations when working with probabilities, especially in selection problems like this one!

There are 14, 12 and 16 students in group A, group B and group C respectively. If 3 students are randomly selected from the 3 groups, find the probability that they come from the same group.

Learn how to calculate the probability of randomly selecting 3 students from the same group when there are multiple groups with different numbers of students. This solution covers key concepts in probability and combinations, including the combination formula and probability calculations.

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