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Question

Two decks of cards are there. Each deck contains 20 cards, with numbers from 1 to 20written on them. A card is drawn of random from each deck, getting the numbers x and y what is the probability that logx+logy is a positive integer. Logs are taken to the base 10.


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Solution

Step 1: Finding the sample space:

Sample space for drawing two cards from each deck is

(1,1),(1,2),(1,3),(1,4),(1,5),.................,(1,20),(2,1),(2,2),(2,3),(2,4),(2,5),.................,(2,20),.................................................................................................................................................................................................................(20,1),(20,2),(20,3),(20,4),(20,5),............(20,20)

Total number of outcomes =400

Step 2: Finding the favorable outcomes:

Since, logx+logy=log(xy)

log10xy is a positive integer if xy=10,100,1000.

i.e.log1010=1;log10100=2;log101000=3

logxy is a positive integer when (x,y)=(1,10),(10,1),(10,10),(5,20),(20,5),(2,5),(5,2)

Number of favorable outcomes =7

Step 3: Finding the probability:

We know, Probability of an event =NumberoffavorableoutcomesTotalnumberofoutcomes

The required probability =7400

Thus, the probability that logx+logy is a positive integer is 7400.


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