There are $\mathrm{400}$ identical seats to be distributed among $\mathrm{3}$ parties such that each party has at most $\mathrm{200}$ seats

Question

There are $\mathrm{400}$ identical seats to be distributed among $\mathrm{3}$ parties such that each party has at most $\mathrm{200}$ seats. What is the number $\mathrm{L}$ of all the possibilities among the above?

There are $\mathrm{{399}\choose{2}}$ possibilities in total and I have to subtract $\mathrm{{198}\choose{2}}$ choices, where there is more than $\mathrm{200}$ seats assigned to one party. Is this answer correct?

Are "seats" distinguishable people? So this is equivalent to saying "There are 400 people to be assigned to three rooms such that...."? Or am I overthinking and you literally mean 400 identical chairs? — AlgTop1854, Oct 14 '23 at 13:08
$~x_1 + x_2 + x_3 = 400~$ and $~x_1,x_2,x_3 \in \Bbb{Z_{0 \leq x_i \leq 200}},~$ with possible additional constraint that each $~x_i \geq 1.~$ See this answer for a blueprint of how to combine Inclusion-Exclusion with Stars-And-Bars to attack this generic type of problem. — user2661923, Oct 14 '23 at 13:41

true blue anil · Accepted Answer · 2023-10-14T14:38:15.480

You have assumed that every party gets at least one seat, which has not been specified in the problem statement.

Also, you have forgotten that any of the three parties must not get more than $200$ seats.

If I go by the wording of the question, the answer using stars and bars (written in the full form) gives

$\dbinom{400+3-1} 2 - 3*\dbinom{(400-201)+3-1}2 = 20,301$

which tallies with David G. Stork's answer

But it is quite possible (though not mentioned) that every party is supposed to get at least $1$ seat.

Can you now work out the answer for this problem (which you were attempting) correctly ? Will it be the same or different ?

David G. Stork · Answer 2 · 2023-10-14T14:34:51.840

0

$$\left( \sum\limits_{i=0}^{200} \sum\limits_{j=200-i}^{200} 1 \right) = 20,301$$

Where $i$ is the number of seats for the first group, $j$ the number in the second group (and the rest are in the third group).

It is unclear whether the groups are considered distinct.

edited Oct 14 '23 at 14:34

answered Oct 14 '23 at 13:30

David G. Stork

29,774

May I ask what's the idea behind this formula? – J P Oct 14 '23 at 13:41

There are $\mathrm{400}$ identical seats to be distributed among $\mathrm{3}$ parties such that each party has at most $\mathrm{200}$ seats

2 Answers2