Joe
Reputation
702
Top tag
Next privilege 1,000 Rep.
Create tags
 Jul2 awarded Curious May1 awarded Yearling Jan29 awarded Notable Question May24 awarded Popular Question May1 awarded Yearling Jan8 asked Can a Hermitian operator on a tensor product space be represented as a sum of tensor products of Hermitian operators? Jan3 awarded Good Answer Jan2 awarded Nice Answer Jan1 awarded Yearling Jan1 answered Are there real-life relations which are symmetric and reflexive but not transitive? Aug5 comment What is the sum of only half the exponential terms that give the Dirac comb? Got it - thanks! Aug5 accepted What is the sum of only half the exponential terms that give the Dirac comb? Aug5 asked What is the sum of only half the exponential terms that give the Dirac comb? Jun8 awarded Caucus Mar7 accepted How many ways can $n$ adults, $k_1$ boys and $k_2$ girls be seated in a line such that no two children of the same sex sit next to each other? Mar7 comment How many ways can $n$ adults, $k_1$ boys and $k_2$ girls be seated in a line such that no two children of the same sex sit next to each other? @dtldarek: why don't you edit your answer to include joriki's correction, and I'll accept it? Mar7 comment How many ways can $n$ adults, $k_1$ boys and $k_2$ girls be seated in a line such that no two children of the same sex sit next to each other? As @joriki pointed out, I did have a mistake in my first comment. The expression I gave for $X_m$ is incorrect, but I believe joriki's expression is correct. Mar7 comment How many ways can $n$ adults, $k_1$ boys and $k_2$ girls be seated in a line such that no two children of the same sex sit next to each other? Nice! This is a similar approach to @dtldarek's answer, only simpler. Though, after joriki's correction that answer becomes simple as well. Mar7 comment How many ways can $n$ adults, $k_1$ boys and $k_2$ girls be seated in a line such that no two children of the same sex sit next to each other? But I think you can put a boy in the left most place in line. If there is an adult there he is next to the adult, and if there is no adult then there is a pair, and he is next to a girl. In both cases it is possible. Also, I don't see how a binomial can ever be equal to zero, since it is defined as a positive integer. Mar7 comment How many ways can $n$ adults, $k_1$ boys and $k_2$ girls be seated in a line such that no two children of the same sex sit next to each other? This approach is a great idea! I think your formula needs some corrections though. First of all, swapping $a$ and $b$ shouldn't matter, so we can assume WLOG $a \ge b$. Second of all, since we can always put a boy in the left most place in the line (whether there's an adult there or not), I think the expression for $X_m$ is simply: $\binom{n+1}{a-m}(n+m)!$. Third, It's not hard to see that there is a minimal number of pairs which might be greater than zero, so the summation should be $\sum_{m=\max{(0,a-n-1)}}^b$.