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# Probability

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Intern
Joined: 12 Oct 2017
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12 Oct 2017, 21:40
12 marbles are selected at random from a large collection of white, red, green and yellow Marbles. The number of Marbles of each colour is unlimited. Find the probability that the selection contains at least one marble of each colour?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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GMAT 1: 760 Q51 V42
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17 Oct 2017, 08:58
manu1 wrote:
12 marbles are selected at random from a large collection of white, red, green and yellow Marbles. The number of Marbles of each colour is unlimited. Find the probability that the selection contains at least one marble of each colour?

Sent from my FRD-L02 using GMAT Club Forum mobile app

Total number of cases is 4^12
The number of cases that only 1 color is chosen 12 times is 4C1 * 1^12 = 4, since we have 4 colors.
The number of cases that only 2 colors of marbles are chosen 12 times is 4C2 * ( 2^12 - 2 ) = 6 * ( 2^12 - 2 ) = 6*2^12 - 12.

The number of cases that only 3 colors of marbles are chosen 12 times is 4C3 * ( 3^12 - 3C2*(2^12-2C1*1^12 ) - 3C1*1^12 ) = 4*(3^12 - 3*(2^12-2) - 3 ) = 4*3^12 - 12*2^12 + 12

When we subtract the sum of the above numbers from 4^12 , we have the number of cases the selection contains at least one marble of each color, which is 4^12 - ( 4*3^12 - 12*2^12 + 12 + 6*2^12 - 12 + 4 ) = 4^12 - ( 4*3^12 - 6**2^12 + 4 ) = 4^12 - 4*3^12 + 6*2^12 - 4
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Re: Probability &nbs [#permalink] 17 Oct 2017, 08:58
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# Probability

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