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 objects are selected and Z represents number of ways to select them such that chosen objects are not diametrically opposite. Which of the following is/are comect?

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 objects are selected and Z represents number of ways to select them such that chosen objects are not diametrically opposite. Which of the following is/are comect? (A) X=1140  (B) Y=800  (C) Z=960  (D) Y ∩ Z=660

Tardigrade Admin · Accepted Answer

X = 20 C 3=1140 Y =(20/3) × 16 C 2=(20 × 16 × 5/6)=800 Z = 20 C 3-10 × 18=1140-180=960 Y ∩ Z = Y text and Z text both = Y -10 × 14=800-140=660

$X = 1140$

$Y = 800$

$Z = 960$

$Y \cap Z = 660$

$X =^{20} C_{3} = 1140$
$Y = \frac{20}{3} \times^{16} C_{2} = \frac{20 \times 16 \times 5}{6} = 800$
$Z =^{20} C_{3} - 10 \times 18 = 1140 - 180 = 960$
$Y \cap Z = Y and Z both = Y - 10 \times 14 = 800 - 140 = 660$

X=1140

Y=800

Z=960

Y∩Z=660

X=20C3​=1140 Y=320​×16C2​=620×16×5​=800 Z=20C3​−10×18=1140−180=960 Y∩Z=Y and Z both =Y−10×14=800−140=660

$X = 1140$

$Y = 800$

$Z = 960$

$Y \cap Z = 660$

$X =^{20} C_{3} = 1140$
$Y = \frac{20}{3} \times^{16} C_{2} = \frac{20 \times 16 \times 5}{6} = 800$
$Z =^{20} C_{3} - 10 \times 18 = 1140 - 180 = 960$
$Y \cap Z = Y and Z both = Y - 10 \times 14 = 800 - 140 = 660$