6 people are playing Secret Santa for Christmas. They will each give one gift to someone, and each receive one gift from someone. They are not allowed to give their gift to themselves. How many different ways are there to do so?
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Let's think of the question this way:
Suppose you sit the 6 people around a round table.
Then everybody gives a present to the person on their left.
So everybody has given a present and received a present.
If we change the "arrangement" of the 6 people around the table, we have a different Secret Santa arrangement.
So all we have to do is calculate the number of ways we can sit 6 people around a round table, (or let them form a circle)
that would be 6!/6 = 5! = 120
(I divided by 6. Here is why. If everybody got up and moved one to the right, the order would not change, and we can do that 6 times)
no, its wrong
To solve this problem, we can use a counting technique called derangements. A derangement is a permutation of a set such that no element appears in its original position.
Let's break down the problem step by step:
1. Consider one person, let's call them Person A. Person A can choose a gift from any of the remaining 5 players (not including themselves). So, there are 5 choices for Person A.
2. After Person A has chosen a gift, there are 4 remaining people left to choose from. Let's consider the next person, Person B. Person B cannot choose themselves or the person who already received a gift from Person A. So, there are 3 choices for Person B.
3. Continuing this pattern, let's consider Person C. Person C cannot choose themselves, Person A, or Person B. So, there are 2 choices for Person C.
4. Moving to Person D, they cannot choose themselves, Person A, Person B, or Person C. Only 1 choice remains for Person D.
5. Now, consider Person E. Person E has no restrictions since all the other 4 people have already received their gifts. So, there is only 1 choice for Person E.
6. Finally, we have Person F. They only have one choice left, which is for themselves since all the other participants have already been assigned gifts.
To find the total number of ways to do Secret Santa, we just multiply the number of choices for each person:
5 × 3 × 2 × 1 × 1 × 1 = 30
Therefore, there are 30 different ways for the 6 people to exchange gifts in a Secret Santa game while following the rules.