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Combinatorics and The Domino Effect

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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Combinatorics and The Domino Effect

by Kyan » Thu Nov 15, 2012 3:25 pm
Question guys, I was reading on the topic of Combinatorics in the Word Translations MGMAT guide.

I had a hypothetical question I was hoping someone can help with. It's a revised version of an example question in the MGMAT Word Translation guide (pg 190).

A miniature gumball machine contains 7 blue, 5 green and 4 red gumballs, which are identical except for their colors. If the machine dispenses three gumballs at random what is the probability that it dispenses 2 green gumballs and 1 blue gumball?

What would be the approach to this?

I'm getting 1/8. This is correct??

Thanks!!
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by David Chong » Thu Nov 29, 2012 12:41 pm
Kyan:

Your answer is correct. Here's how I figure it--

Just as in the original example on page 190, we take the individual probabilities of each draw (in this case GREEN, then GREEN, then BLUE) and multiply them together.
  • (5/16) * (4/15) * (7/14) = 1/24
Notice that for the probability of getting a green gumball on the second draw (bolded above), both the total number of gumballs in the denominator and the remaining number of green gumballs in the numerator decreased by one, reflecting the fact that we already received a green gumball on our first draw.

So 1/24, which is the probability of drawing GREEN GREEN BLUE, must now be multiplied by the number of ways that {G G B} can be reordered. You can either figure this out by listing the possibilities,
  • GGB
    GBG
    BGG
or by calculating a permutation of 3 items, 2 of which are identical (check out page 71 for more on this!):
  • 3!/2! = 3
Either way,
  • (1/24) * 3 = 1/8
I'm curious to hear whether your process differed substantially from this one, since we both arrived at the same value. Hope this was helpful!


--
David Chong
Manhattan GMAT
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