In order to conduct an​ experiment, 5 subjects are randomly selected from a group of 51 subjects. How many different groups of 5 subjects are​ possible?


To determine how many different groups of 5 subjects can be selected from a group of 51 subjects, we use the concept of **combinations**. Since the order in which the subjects are selected doesn't matter, this is a combination problem.

The formula for combinations is:

\[
C(n, r) = \frac{n!}{r!(n - r)!}
\]

Where:
- \(n\) is the total number of subjects (in this case, 51),
- \(r\) is the number of subjects to be selected (in this case, 5).

Now, applying the formula:

\[
C(51, 5) = \frac{51!}{5!(51 - 5)!} = \frac{51!}{5!46!}
\]

We can calculate this step-by-step:

\[
C(51, 5) = \frac{51 \times 50 \times 49 \times 48 \times 47}{5 \times 4 \times 3 \times 2 \times 1} = \frac{254251200}{120} = 2118760
\]

So, there are **2,118,760** different groups of 5 subjects possible.

Would you like more details or have any questions? Here are some related problems:

1. How many ways can 3 subjects be selected from a group of 10?
2. What is the difference between combinations and permutations?
3. How would the result change if the order of selection mattered?
4. How many ways can 5 subjects be selected from a group of 20?
5. How can combinations be applied to real-world scenarios outside experiments?

**Tip:** When dealing with combinations, always check if the order matters; if it does, it's a permutation, not a combination!

In order to conduct an experiment, 5 subjects are randomly selected from a group of 51 subjects. How many different groups of 5 subjects are possible?

Learn how to calculate the number of ways to select 5 subjects from a group of 51 using combinations. This problem involves the combination formula C(n, r) = n! / (r!(n - r)!) and is suitable for students in Grades 9-12.

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