Line up the eight people in some order, say alphabetically. Hand each of them a chair. That leaves $22$ chairs. Line up those $22$ chairs in a row. This creates $23$ spaces, $21$ between successive chairs and $2$ at the ends of the row, in which a person can insert a chair. To ensure that the people are separated, each person must place his or her chair in a separate space. The first person has $23$ options, which leaves $22$ options for the next person, and so forth. Hence, the number of permissible seating arrangements is
$$23 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 = 23 \cdot 22 \cdot 21 \cdot 20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot \frac{15!}{15!} = \frac{23!}{15!}$$
Your second answer is correct.
In how many ways can six boys and four girls be seated around a table if no two girls are adjacent?
Method 1: Suppose one of the boys is Asa. We will use him as a reference point. The remaining five boys can be seated in $5!$ ways as we proceed clockwise around the table from Asa. Seating the six boys creates six spaces in which a girl could be placed, one to the left of each boy. To separate the girls, we must choose four of these six spaces in which to place a girl. The girls can be arranged in the four selected spaces in $4!$ ways as we proceed clockwise around the table from Asa. Hence, the number of admissible seating arrangements is
$$5!\binom{6}{4}4!$$
Method 2: We modify your attempt. Suppose Adrienne is one of the girls. The other girls can be seated in $3!$ ways as we proceed clockwise around the table from Adrienne. This creates four spaces in which to place the boys, one to the left of each girl. Let $x_1, x_2, x_3, x_4$ denote, respectively, the number of boys in the first, second, third, and fourth spaces as we proceed clockwise around the table from Adrienne. Since there are a total of six boys,
$$x_1 + x_2 + x_3 + x_4 = 6$$
Since no two of the girls are adjacent, there must be at least one boy in each of the four spaces. Hence, this is an equation in the positive integers. A particular solution of the equation corresponds to placing three addition signs in the five spaces between successive ones in a row of six ones.
$$1 \square 1 \square 1 \square 1 \square 1 \square 1$$
For instance, placing an addition sign in the first, second, and fourth spaces gives
$$1 + 1 + 1 1 + 1 1$$
which corresponds to the solution $x_1 = 1$, $x_2 = 1$, $x_3 = 2$, $x_4 = 2$. The number of such solutions is the number of ways we can place three addition signs in the five spaces between successive one in a row of six ones, which is
$$\binom{6 - 1}{4 - 1} = \binom{5}{3}$$
One the number of boys in each space has been selected, the boys can be arranged in those spaces in $6!$ ways as we proceed clockwise around the table from Adrienne. Hence, there are
$$3!\binom{5}{3}6!$$
admissible seating arrangements.
Method 3: We correct your approach. Again, suppose Adrienne is one of the girls. The other girls can be seated in $3!$ ways as we proceed clockwise around the table from Adrienne. Choose which four of the six boys will sit to the immediate left of a girl. Those boys can be arranged in $4!$ ways as we proceed clockwise around the table from Adrienne. That leaves two boys to place. There are two possibilities: both boys are placed between the same two girls so that there are three boys between those girls or they are placed between separate pairs of girls so that there are two pairs of girls with two boys between them.
Both of the remaining boys are placed between the same two girls: There are four ways to choose the pair of girls the boys sit between. Both boys must sit to the left of the boy who has already been seated to the immediate left of the first girl we encounter in the pair as we proceed clockwise around the table from Adrienne (starting with Adrienne). There are two ways to choose which of the boys who has not yet been seated sits next to that boy. Hence, there are $\binom{4}{1}2!$ such cases.
The remaining boys are placed between two different pairs of girls: There are $\binom{4}{2}$ ways to choose which two pairs of girls the boys sit between. In each case, the boy must be seated to the left of the boy who has already been seated to the immediate left of the first girl we encounter in the pair as we proceed clockwise around the table from Adrienne (starting with Adrienne). There are two ways to choose which of the boys we will encounter first as we proceed clockwise around the table from Adrienne. There are $\binom{4}{2}2!$ such cases.
Thus, there are
$$3!\binom{6}{4}4!\left[\binom{4}{1}2! + \binom{4}{2}2!\right]$$
admissible seating arrangements.
Your errors:
You forgot to arrange the four boys you placed in the gaps, so you were missing a factor of $4!$. Had you included that factor, your count would have been too large. The reason for this is that you counted the same arrangement more than once. For instance, if Asa, Bradley, and Charles all sit between Adrienne and Bronwyn, your approach would count the same arrangement $3! = 6$ times, corresponding to the $3!$ orders in which the same boys could be placed in the same seats. If Asa and Bradley were to be placed between Adrienne and Bronwyn and Charles and David were to be placed between Bronwyn and Claire, then your approach would count the same arrangement four times, corresponding to the $2!$ orders in which Asa and Bradley could be placed in the same seats and the $2!$ orders in which Charles and David could be placed in the same seats.
Best Answer
The original problem could be more clearly worded, I think.
When you say in Method 1 that there are 6 possible choices for person 1, this is "true", but only if you actually care about the seats as being distinct seats, as opposed to "arrangements of people" being what we care about. The key difference is that in this problem where there's a "circular table" in which a rotation of the table doesn't change the "arrangement of people".
For example, let's say that there are 6 chairs placed around a clock, just at the even numbers. So a chair at 2:00, at 4:00, ..., at 12:00. Suppose that you've placed all your people around the clock. Great! There's an arrangement! And your current solution keeps these clock numbers in mind, and this is why you get the answer you do.
But now rotate the clock so that 2:00 becomes 4:00, that 4:00 becomes 6:00, etc. The arrangement you found has now changed as long as we keep track of these hour markings. In fact, there are $6$ ways for you to "rotate" the clock around, keeping everyone in the exact same position (rotate by 2 hours, by 4 hours, by 6 hours,..., by 12 = 0 hours).
Now, in the table problem, we have no hour markings on our table, and so all of these rotations shouldn't affect what an arrangement of people looks like!
All that to say, you have overcounted by a factor of $6$. That is, once you pick a configuration of people around this clock, we remember that we aren't sitting around a clock, we're around a table without numbers to tell us "which hour" someone is sitting at. So all of the rotations we got beforehand should actually be "the same".
What we get, then, is a $6$-to-$1$ function $\{\text{clock arrangements}\} \to \{\text{table arrangements}\}$ where the function takes our clock arrangement and "forgets" the numbers on the clock. (Double check that there are always $6$ "clock" arrangements that correspond to every "table" arrangement.) We know (I hope) that if $f : A \to B$ is an $n$-to-$1$ function, then $|A| = |B|\cdot n$, or equivalently $|B| = \frac{|A|}{n}$, and so the $6$-to-$1$ function we have described just means we need to divide the current answer by $6$. Hence, $\frac{432}{6} = 72$.