A box contains card labeled 1 to 20. What is the probability that the product of any three cards picked up is a multiple of 21.  

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justaguide | College Teacher | (Level 2) Distinguished Educator

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There are cards from 1 to 20 in the box. 3 cards are picked at random. We need to determine the probability the product of the three cards is a multiple of 21.

Now, to get a multiple of 21 we need one card which is a multiple of 3 and another which is a multiple of 7, the third can be any card.

Out of the numbers from 1 to 20, we have 6 numbers that are multiples of 3 and 2 numbers that are a multiple of 7.

If we denote numbers that are divisible by 3 as T, those divisible by 7 as S, and the others as X, the cards can be picked up in any of the following orders: TSX , TXS, XTS, XST, STX and SXT.

The probability of picking up a card with a number divisible by 3 is 6/20, the probability of picking up a card with a number divisible by 7 is 2/20 and the probability of any other number is 18/20.

This gives a probability of (6/20)*(2/20)*(18/20)*6 = .162

Therefore the required probability is 0.162

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