You have 6 different New Year greeting cards and 4 friends. If each friend is to receive exactly one card, and you choose 4 of the 6 cards to send (one distinct card to each friend), in how many different ways can this be done?

Introduction / Context:
This question is about distributing distinct objects to distinct recipients under the condition that each recipient receives exactly one object. We have 6 different greeting cards and 4 friends, and we send one card to each of the 4 friends, using 4 of the 6 cards.

Given Data / Assumptions:

• Number of distinct greeting cards = 6.
• Number of friends = 4.
• Each friend must receive exactly one card.
• No friend receives more than one card.
• Cards are distinct and friends are distinct.

Concept / Approach:
We interpret the process in two steps. First, we select which 4 cards (out of 6) will be used. Second, we arrange those 4 selected cards among the 4 friends. Because order of assignment matters, we must count permutations rather than just combinations.

Step-by-Step Solution:
Step 1: Choose 4 cards from the 6 distinct cards. Number of ways = C(6, 4). Step 2: Evaluate C(6, 4) = 6 * 5 / (2 * 1) = 15. Step 3: For any fixed set of 4 cards, assign them to 4 friends. The number of ways to assign 4 distinct cards to 4 distinct friends is 4! = 24. Step 4: Total ways = number of ways to choose cards * number of ways to assign them = 15 * 24. Step 5: Compute 15 * 24 = 360.

Verification / Alternative check:
Another way is to view the process directly as permutations: we are creating an ordered selection of 4 cards from 6 for the 4 friends. This is 6P4 = 6 * 5 * 4 * 3, which equals 360 as well, confirming the two step reasoning.

Why Other Options Are Wrong:
720 would correspond to 6!, which would be appropriate only if all 6 cards were used and arranged. 240 and 740 do not match the natural structure of 6P4 or C(6, 4) * 4! and therefore are miscalculations or guesses.

Common Pitfalls:
Students sometimes use only combinations and stop at C(6, 4) = 15, forgetting that each chosen set can be arranged among the friends in many ways. Others mistakenly use 6^4, which would allow a card to be sent to more than one friend, violating the condition that each card is used at most once.

Final Answer:
The number of ways to send 4 different cards to 4 friends, choosing from 6 available cards, is 360.