From a group of 4 people, in how many different ways can ordered arrangements of 3 people be formed (that is, in how many ways can we select groups of 3 at a time where order matters)?

Introduction / Context:
This question is directly about permutations. We have 4 distinct people and want to know in how many ways we can form ordered groups of 3 people. Because the order within each group matters, different sequences of the same 3 people are treated as different arrangements. This is a classic example of permutations of n distinct items taken r at a time.

Given Data / Assumptions:

• Total people available = 4.
• We form groups of size 3.
• Within each group, order matters (for example, first, second and third positions are distinct).
• No person is repeated in the same arrangement.

Concept / Approach:
The number of ordered arrangements of r items from n distinct items is given by the permutation formula P(n, r) = n! / (n - r)!. Here, n = 4 and r = 3. We can apply this formula directly or count the choices position by position, multiplying the number of choices for each position.

Step-by-Step Solution:
Step 1: Use the permutation formula. P(4, 3) = 4! / (4 - 3)! = 4! / 1!. 4! = 4 * 3 * 2 * 1 = 24. So, P(4, 3) = 24. Step 2: Alternatively, count position by position. For the first position, there are 4 choices. For the second position, after one person is chosen, 3 people remain. For the third position, after two people are chosen, 2 people remain. Multiplying, number of ways = 4 * 3 * 2 = 24.

Verification / Alternative check:
Listing all permutations explicitly would confirm that there are 24 distinct ordered triples. However, this is not necessary once we realise that the permutation formula and the direct positional counting both agree on 24. Both methods are logically equivalent and consistent with basic permutation principles.

Why Other Options Are Wrong:
The value 16 would correspond to 4^2, which is not relevant here. The number 20 might be mistakenly obtained by using combinations (C(4, 3) = 4) and then multiplying incorrectly. The value 36 is larger than 4 * 3 * 2 and would require an extra factor that does not exist in this setting. The value 12 is exactly half of the correct answer and could result from stopping the multiplication early.

Common Pitfalls:
A common mistake is to use combinations, which count unordered selections, rather than permutations. Another error is to think that there are only C(4, 3) = 4 groups and forget that each group can be arranged in 3! = 6 ways, giving 4 * 6 = 24 ordered arrangements. Keeping the distinction between order sensitive and order insensitive problems clear helps avoid such confusion.

Final Answer:
There are 24 different ordered arrangements of 3 people from the group of 4.