Question

There are two distinct round tables, each with 5 seats. In how many ways may a group of ten people sit?

Answer 1

i am going to go out on a limb and say
\[\dbinom{10}{5}4!4!\] then you can tell me why this is wrong

Answer 2

well i haven't figured out the right answer for this one yet lol. but the answer is 145 152

Answer 3

oh well let me check is this is the same!
GOT IT!

Answer 4

yay

Answer 5

now that i am confident i am not a total moron when it comes to counting, would you like the reasoning?

Answer 6

yes please :)

Answer 7

i reasoned as follows:
the number of ways to put 5 people at one table and 5 at another is the same as the number of ways to choose 5 from a set of 10 which by definition is
\[\dbinom{10}{5}\]

Answer 8

and
\[\dbinom{10}{5}=\frac{10\times 9\times 8\times 7\times 6}{5\times 4\times 3\times 2}=2\times 9\times 2\times 7=252\]

Answer 9

yep

Answer 10

then once you have them divided in to two groups, the number of ways to arrange 5 people in a circle is 4! and since there are two tables you have 4! for one and 4! for the other so "counting principle" says multiply all this together

Answer 11

btw there is a reasonable explanation for why if you have n people in a circle there are (n-1)! ways to arrange them, whereas if you have n people in a line there are n! ways

Answer 12

and there is even a nice picture. if you want you can read the first paragraph here, which says it a lot better than i can
http://www.learner.org/courses/mathilluminated/units/2/textbook/05.php
first three pictures really says it all

Answer 13

oh wow:) thanks a lot for your help