How many ways are there to divide a group of 6 friends among the basketball team, the soccer team, and the track team?  (Each team could have anywhere from 0 to 6 of the friends on it.  Assume the friends are distinguishable.)
For each friend, there are 3 options for which team to put them on. Since each of the 6 people have 3 options, there are $3^6=\boxed{729}$ to divide the group of friends.
729
Level 5
Counting & Probability
Problem: How many ways are there to divide a group of 6 friends among the basketball team, the soccer team, and the track team?  (Each team could have anywhere from 0 to 6 of the friends on it.  Assume the friends are distinguishable.)
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the number of ways to assign 6 distinguishable friends to 3 different teams, we can use the principle of counting. Each friend has 3 independent choices: they can join the basketball team, the soccer team, or the track team. 

For each of the 6 friends, the choices can be represented as follows:

- Friend 1: 3 choices (basketball, soccer, track)
- Friend 2: 3 choices (basketball, soccer, track)
- Friend 3: 3 choices (basketball, soccer, track)
- Friend 4: 3 choices (basketball, soccer, track)
- Friend 5: 3 choices (basketball, soccer, track)
- Friend 6: 3 choices (basketball, soccer, track)

Since the choices for each friend are independent, we can multiply the number of choices for each friend together. This leads to the calculation:

Total ways = 3 (choices for Friend 1) × 3 (choices for Friend 2) × 3 (choices for Friend 3) × 3 (choices for Friend 4) × 3 (choices for Friend 5) × 3 (choices for Friend 6)

This can be expressed mathematically as:

Total ways = 3^6

Calculating 3^6 gives us:

3^6 = 729

Thus, the total number of ways to divide the group of 6 friends among the basketball team, the soccer team, and the track team is 729.<|END_OF_LATENT|> The final answer is $729$.