In how many different ways can you split $10$ people into two groups of $5$ people?

If the groups are labeled, your answer is correct. Otherwise, notice that choosing five people to be in the first group automatically determines who is in the second group. When you count combinations of five of the ten people, you count each group twice, once when you select the group and once when you select its complement. Therefore, unless the groups are labeled, the number of ways to split ten people into two groups of five is $$\frac{1}{2}\binom{10}{5}$$

Edit: @GlenO proposed the following alternative argument in the comments: Suppose Alicia is one of the ten people. The number of ways two groups of five people can be formed is the number of ways of selecting four of the remaining nine people to be in Alicia's group or, equivalently, the number of ways of selecting five of the remaining nine people to not be in Alicia's group. Therefore, the number of ways to split two people into two groups of five is $$\binom{9}{4} = \binom{9}{5}$$ which is equivalent to the answer I obtained above.