Answer to Question #325618 in Discrete Mathematics for shadi

Question

Number plates in an Arab country consist of 3 letters (chosen from a specified set of 20 letters from the Arabic alphabet) and 4 digits (from 0 to 9).

⦁ Find the probability of a plate having identical letters.

⦁ Find the probability of a plate having identical digits.

⦁ Find the probability of a plate having consecutive digits.

Answer 1

$1:\\20\cdot \frac{1}{20}\cdot \frac{1}{20}\cdot \frac{1}{20}=0.0025\\2:\\10\cdot \frac{1}{10}\cdot \frac{1}{10}\cdot \frac{1}{10}\cdot \frac{1}{10}=0.001\\3:\\The\,\,4 digits\,\,in\,\,ascending\,\,order\,\,can\,\,start\,\,from\,\,0,1,2,3,4,5\\That\,\,is\,\,6 ways\\There\,\,are\,\,2 ways\,\,to\,\,choose\,\,order\,\,\left( ascending\,\,or\,\,descending \right) \\Total\,\,6\cdot 2=12 ways\,\,of\,\,10^4=10000 variants\,\,of\,\,digits\\The\,\,probability\,\,is\\\frac{12}{10000}=0.0012$