# Number of club members if they are organized into four committees if each member is in exactly two committees and any two committees share one member

A club with $x$ members is organized into four committees such that

$(a)$ each member is in exactly two committees,

$(b)$ any two committees have exactly one member in common.

Then $x$ has

$(A)$ exactly two values both between $4$ and $8$

$(B)$ exactly one value and this lies between $4$ and $8$

$(C)$ exactly two values both between $8$ and $16$

$(D)$ exactly one value and this lies between $8$ and $16$.

My thought is option $B$ and $x = 6$. Please somebody tell me am I right or wrong?

• Can you show how you arrived at $x=6$? – RandomUser Apr 25 '14 at 16:53
• I just see that if $x$=$6$ then the two conditions are satisfied.But if $x$<$6$ or $x$>$6$ the condition $b$ is not satisfied so I think $x$ may be $6$. – liesel Apr 25 '14 at 17:15
• Please improve your title. – Alexander Gruber Apr 25 '14 at 17:22

Claim: $x=6$

Let $S:=[x]\left(=\{1,2,\dots ,x\}\right)$ where $x$ is some fixed natural number. Let $C_i\subset S$ be the $i$ th commitee.Let $f$ be a function from the set of unordered pair of distinct commitees to members of the club defined as, $$f\left( (C_i,C_j)\right)=C_i\cap C_j$$ $f$ is injective by (a). $f$ is surjective by (b).

Hence, $|S|=\binom{4}{2}=6$.

Here is a construction:

• Let $C_1\cap C_2=1$ wlog. Then, $1\notin C_i \quad \forall \, i\in [4]\setminus \{1,2\}$.

• Let $C_2\cap C_3=2$ wlog. Then, $2\notin C_i \quad \forall \, i\in [4]\setminus \{2,3\}$.

• Let $C_3\cap C_4=3$ wlog. Then, $3\notin C_i \quad \forall \, i\in [4]\setminus \{3,4\}$.

• Let $C_1\cap C_3=4$ wlog. Then, $4\notin C_i \quad \forall \, i\in [4]\setminus \{1,3\}$.

• Let $C_1\cap C_4=5$ wlog. Then, $5\notin C_i \quad \forall \, i\in [4]\setminus \{1,4\}$.

• Let $C_2\cap C_4=6$ wlog. Then, $6\notin C_i \quad \forall \, i\in [4]\setminus \{2,4\}$.

Let the committees be $A_1, A_2, A_3, A_4$ then $|A_1\cup A_2 \cup A_3 \cup A_4|=n$. There are $\binom{4}{2}=6$ pairwise intersections each of size $1$ and each intersection of triples must be empty since if not some member would belong to $3$ committees, a contradiction. If we make a $4 \times n$ table where rows are committees and columns are members and mark a $1$ whenever member $x_j$ belongs to committee $A_i$ we notice that the $i^{th}$ row sum is $|A_i|$ and the $j^{th}$ column sum is $2$ (since each member belongs to exactly $2$ committees) so equating row and column sums we get $\sum_{i=1}^{4}|A_i|= 2n.$

Putting this information together and using the inclusion-exclusion formula we get $n =6$