In how many distinct ways can I arrange my five keys on a keychain, if I want to put my house key next to my car key? Two arrangements are not considered different if the keys are in the same order (or can be made to be in the same order without taking the keys off the chain--that is, by reflection or rotation).
First, put the house key and car key next to each other on the keychain. It doesn't matter where they are put, because the keychain can be rotated and reflected to move them to any other pair of adjacent locations. The remaining three keys can be put on the keychain in $3!=\boxed{6}$ ways.
6
Level 4
Counting & Probability
Problem: In how many distinct ways can I arrange my five keys on a keychain, if I want to put my house key next to my car key? Two arrangements are not considered different if the keys are in the same order (or can be made to be in the same order without taking the keys off the chain--that is, by reflection or rotation).
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the problem, first, consider the house key and car key as a single unit or block since they must be next to each other. This block can be arranged in two ways: house key first or car key first. 

Next, treat this block as one key, reducing the total number of keys from five to four: the house-car block, and the three other keys. 

Now, calculate the arrangements of these four units on a keychain. The formula for arranging n distinct objects in a circle is (n-1)!. For four units, the number of arrangements is (4-1)! = 3! = 6.

Since the house-car block can be arranged in 2 ways, multiply the circular arrangements by 2. Thus, the total number of distinct arrangements is 6 * 2 = 12.

However, since the problem states that arrangements are not considered different if they can be made to be in the same order by reflection or rotation, we need to account for this symmetry. 

In a circular arrangement, each arrangement can be reflected, effectively halving the total distinct arrangements. Therefore, the final count of distinct arrangements is 12 / 2 = 6. 

Thus, the final answer is 6.<|END_OF_LATENT|> The final answer is $6$.