On the occasion of New Year, each of the 21 students in a class sends a greeting card to every other student in the class. What is the total number of greeting cards exchanged among the students?

Introduction / Context:
This is a classic counting problem involving pairwise exchanges between students. Each student sends greeting cards to all others, so we must count the total number of directed cards (from sender to receiver). It tests the understanding of how many ordered pairs of distinct students can be formed from a given group size. This kind of thinking appears in graph theory (directed edges), handshake problems, and simple combinatorial reasoning.

Given Data / Assumptions:
- Total number of students in the class: 21.
- Each student sends a greeting card to every other student.
- No student sends a card to themselves.
- Cards are directed: a card from student A to student B is distinct from a card from B to A.

Concept / Approach:
For each student, count how many cards they send, then sum over all students. Since every student has the same number of recipients, we can multiply. Each student sends a card to every other student, which means to 21 - 1 = 20 other students. There are 21 students in total, so the total number of cards is 21 multiplied by 20. This directly uses the multiplication principle: (number of possible senders) * (number of possible receivers for each sender).

Step-by-Step Solution:
Step 1: Fix one student as the sender.Step 2: That student sends greeting cards to all remaining 20 students in the class.Step 3: So each student sends 20 cards.Step 4: There are 21 students acting as senders.Step 5: Multiply to get the total number of cards: 21 * 20.Step 6: Compute 21 * 20 = 420.

Verification / Alternative check:
We can also think in terms of ordered pairs (sender, receiver). The number of ordered pairs of distinct students from a set of 21 is 21 * 20, because there are 21 choices for the sender and then 20 remaining choices for the receiver. This is exactly the same calculation as above. If the problem had asked for undirected exchanges (counting a pair of students only once, regardless of who sends to whom), we would divide by 2, but here each direction is a separate card, so we keep 21 * 20.

Why Other Options Are Wrong:
- 380 and 400 are less than 420 and could arise from subtracting some pairs or miscounting the number of recipients per student.

- 441 is 21^2 and would incorrectly include pairs where a student sends a card to themselves, which is not allowed in this scenario.

Common Pitfalls:
Some learners mix up directed and undirected counting. If one thinks only in terms of pairs of students exchanging cards, they might compute 21C2, which gives 210, and then double it or forget to double it. It is important to interpret that “each student sends a card to every other student” means we care about direction: A to B and B to A are two separate cards. Keeping the distinction between ordered and unordered pairs in mind avoids confusion.

Final Answer:
The total number of greeting cards exchanged among the 21 students is 420.