Question 827728
<pre>
{{{(matrix(17,1,

Number, of, ways, 10, people, can, sit, if, person, 3, "can't", sit, between, person, 1, and, 2))}}}  {{{""=""}}}  {{{(matrix(10,1,

Number, of, ways, 10, people, can, sit, in, any, seat))}}}{{{""-""}}}{{{(matrix(25,1,

Number, of, ways, 10, people, can, sit, if, person, 3, sits, between, persons, 1, and, 2,with,person,1, on,the,LEFT,of,person,3))}}}{{{""-""}}}{{{(matrix(25,1,

Number, of, ways, 10, people, can, sit, if, person, 3, sits, between, persons, 1, and, 2,with,person,1, on,the,RIGHT,of,person,3))}}}


{{{(matrix(10,1,

Number, of, ways, 10, people, can, sit, in, any, seat))}}} {{{""=""}}} {{{10!}}} {{{""=""}}} {{{3628800}}}


{{{(matrix(25,1,

Number, of, ways, 10, people, can, sit, if, person, 3, sits, between, persons, 1, and, 2,with,person,1, on,the,LEFT,of,person,3))}}} {{{""=""}}} {{{8!}}} {{{""=""}}} {{{40320}}}

Explanation:
We consider the "trio" (persons 1,3,2 in that order) as 1 single "thing" and
that there are 8 "things", 7 single people and 1 "trio".  That's 8! ways.

{{{(matrix(25,1,

Number, of, ways, 10, people, can, sit, if, person, 3, sits, between, persons, 1, and, 2,with,person,1, on,the,RIGHT,of,person,3))}}} {{{""=""}}} {{{8!}}} {{{""=""}}} {{{40320}}}

Explanation:
Just like the preceding case, we consider the "trio" (persons 2,3,1 in that
order) as 1 single "thing" and that there are 8 "things", 7 single people and
1 "trio".  That's also 8! or 40320 ways.

Answer:  10! - 2×8! = 3628800 - 2×40320 = 3548160 ways.

Edwin</pre>