Question

There are $20$ points lying on a circle, distance between two consecutive points is same. Three points are selected at random. $X$ represents number of ways to select them. $Y$ represents number of ways to select them such that no two adjacent points are selected and $Z$ represents number of ways to select them such that chosen points are not diametrically opposite. Which of the following is/are CORRECT?

• A. $X=1140$
• B. $Y=800$
• C. $Z=960$
• D. $Y\cap Z=660$

Answer 1

Answer: (A), (B), (C), (D)

$Y\cap Z=660$

$X={}^{20}{C}_3=1140$
$Y=\frac{20}{3}\times {}^{16}{C}_2=\frac{20\times 16\times 15}{6}=800$
There are $10$ diameters
$Z={}^{20}{C}_3-10\times {}^{18}{C}_1=1140-180=960$
$Y\cap Z=Y$ and $Z$ both
$=$ ways in which no two adjacent points are selected $-$ chosen points are not diametrically opposite
$=Y-10\times {}^{14}{C}_1=800-140=660$