Question

A club with $x$ members is organized into four committees in accordance with these two rules.
$\left(1\right)$ Each member belongs to two and only two committees.
$\left(2\right)$ Each pair of committees has one and only one member in common.
Then $x$:

• A. Cannot be determined
• B. Has a single value between $8$ and $16$
• C. Has two values between $8$ and $16$
• D. Has a single value between $4$ and $8$
• E. Has two values between $4$ and $8$

Answer 1

Answer: (D)

Has a single value between $4$ and $8$

This problem may be interpreted as a miniature finite geometry of $4$ lines and $6$ points, so that each pair of lines has only $1$ point in common, and each pair of points has only $1$ line in common;
or
2) $=\frac{4!}{2!2!}=6$, displayed as follows:

Committees	$A$	$B$	$C$	$D$
Members	$a$ $c$ $b$	$b$ $e$ $d$	$d$ $a$ $f$	$f$ $e$ $c$

The number $h$ is known as the height of the polynomial. It is utilized in proving at the number of polynomials with integer coefficients is a countable infinity, the lowest one in the scale of infinities.