From a group of 4 people, in how many different ways can we choose a group of 3 people if the order of people in the group does not matter?

Introduction / Context:
This is a very simple combinations question. We have a small set of people and we must form a smaller group, where the order of selection does not matter. Questions like this introduce the basic concept of combinations C(n,r).

Given Data / Assumptions:

• Total number of people available = 4.
• We must form a group of exactly 3 people.
• Each person is distinct.
• Order of people inside the group is irrelevant.

Concept / Approach:
The number of ways to choose r objects from n distinct objects when order does not matter is given by C(n,r) = n! / (r! * (n - r)!). Here, n = 4 and r = 3. This formula automatically accounts for the fact that different orders of the same people are not counted separately, which is what we want for a committee or group.

Step-by-Step Solution:
We need to compute C(4,3). Using the formula: C(4,3) = 4! / (3! * 1!). Compute 4! = 4 * 3 * 2 * 1 = 24. Compute 3! = 3 * 2 * 1 = 6. Now C(4,3) = 24 / (6 * 1) = 24 / 6 = 4. So there are 4 different groups of 3 people that can be selected.

Verification / Alternative check:
We can list possibilities to confirm. Suppose the four people are A, B, C and D. The possible 3 person groups are: ABC, ABD, ACD and BCD. We can see there are only 4 distinct groups and that this matches the value of C(4,3). This simple listing check is possible because the numbers are very small.

Why Other Options Are Wrong:

• 12: This is 4P3, the number of permutations, and incorrectly treats different orders of the same group as different outcomes.
• 36: This is even larger and corresponds to overcounting by adding extra factors not justified by the problem.
• 16: This figure does not come from any standard nCr or nPr value for n = 4 and r = 3.

Common Pitfalls:
The main pitfall is confusion between permutations and combinations. For groups or committees where order does not matter, combinations must be used. Students sometimes automatically use nPr, which leads to inflated answers. Another issue is not recognising that C(4,3) equals C(4,1), which can sometimes give a quick shortcut. Always check whether position or order is important before choosing the formula.

Final Answer:
The number of different 3 person groups that can be formed from 4 people is 4.