Exercise · ZW-STEINER-TRIPLE-SYSTEM-001 · ★☆☆ Basic
How many pairs of distinct points does a set of \(v\) points contain?
Choose one option.
Show answer
- B. \(\binom{v}{2}\)
Explanation
The number of unordered pairs of \(v\) distinct points is \(\binom{v}{2}=\frac{v(v-1)}{2}\). Counting \(v^2\) confuses pairs with ordered pairs that include repetitions.
Review the concept: Steiner triple system
A Steiner triple system \(\mathrm{STS}(v)\) is a block design in which \(v\) points are grouped into triples so that every pair of distinct points lies in exactly one triple.
\[b=\frac{v(v-1)}{6},\quad r=\frac{v-1}{2}\]
More practice problems: steiner triple system
- ZW-STEINER-TRIPLE-SYSTEM-002 How many pairs does a block of size 3 cover in a Steiner triple system?
- ZW-STEINER-TRIPLE-SYSTEM-003 If v=7, how many blocks does an STS(7) have?
- ZW-STEINER-TRIPLE-SYSTEM-004 How many blocks contain each point in an STS(9)?
- ZW-STEINER-TRIPLE-SYSTEM-005 Which value of v is required for a Steiner triple system to exist?
- ZW-STEINER-TRIPLE-SYSTEM-006 An STS(v) has b=35 blocks. What is the value of v?
- ZW-STEINER-TRIPLE-SYSTEM-007 In an STS(v), how many blocks pass through a fixed point if v=13?
- ZW-STEINER-TRIPLE-SYSTEM-008 How many blocks does an STS(13) have?
- ZW-STEINER-TRIPLE-SYSTEM-009 An STS(v) has r=6 and b=26. Is v=13 the only possible value, and what about v=12? Justify using the…