In the first round, one person chooses 3 people. In round 2, those 3 people choose 3 people each. Keeping with the pattern, how many people get chosen in round 6?

${3}^{6} = 729$

Explanation:

We have six Rounds of choosing people:

R1: 3 are chosen

R2: 3 choose 3, or $3 \times 3 = {3}^{2} = 9$ are chosen

R3: 9 choose 3, or $9 \times 3 = {3}^{2} \times 3 = {3}^{3} = 27$ are chosen

and so on.

And so we can conclude that the pattern of the number of people being chosen in any given round is:

${3}^{R}$, where $R = \text{round}$

Therefore, in the 6th round, we'll have:

${3}^{6} = 729$