A probability problem by prakriti bansal

Probability Level 4

A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let m/n in lowest terms be the probability that no two birch trees are next to one another. Find m+n .


The answer is 106.

Prakriti Bansal by prakriti bansal , 23, China

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1 solution

Dieuler Oliveira
Aug 8, 2014

I) The total of of possible arrangements is: 12 ! 3 ! 4 ! 5 ! = 27720 \frac{12!}{3!4!5!}=27720 .

II) In order to calculate the total of arrangements in which birch trees are not together, first we calculate how many ways we can set maple and oak trees: ( 7 3 ) = 35 {7\choose3} = 35 . Now we have to choose the places between each arrangement and multiply by the total. For example, we got 8 places for the following arrangement: 

                    __O__M__O__M__O__O__M__

Now we just need to choose 5 out of the 8 places available to set the birch trees, that is: ( 8 5 ) = 56 {8\choose 5} = 56 .

The total of possible arrangements are then: 56.35 = 2016. 56.35=2016.

The probability searched is, then: p = 2016 27720 = 7 99 . p=\frac{2016}{27720}=\frac{7}{99}. The answer is: 7 + 99 = 106 . 7+99=\boxed{106}.

4 Helpful 0 Interesting 0 Brilliant 0 Confused

That's the wrong fraction but the right answer

Matt Zhao - 6 years, 2 months ago

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