Coconut Crossing

Probability Level 4

Darby is on a deserted island and wishes to make an $X$ on the beach using $6$ brown coconuts, $9$ green coconuts, her red shoe, and a white frisbee. If each stroke of the $X$ has $9$ objects with one overlap in the center, in how many different ways can this be done?

Details and Assumptions:

The $X$ has strokes which meet at a right angle, so that when viewed correctly it also looks like a plus sign ( $+$ ).
Two configurations are distinct if they cannot be obtained through rotation.

The answer is 340340.

1 solution

Adam Hufstetler
Jun 23, 2017

If we disregard symmetry, this problem becomes a simple combinatorics problem. We must find the number of ways to place the $6$ brown coconuts into $17$ spots, then the number of ways to place $9$ green coconuts in the $11$ spots left, and finally the number of ways to place the $1$ red shoe in the $2$ spots left. After we find these numbers we must multiply them together. Thus we must find:

$\displaystyle \binom{17}{6} \cdot \binom{11}{9} \cdot \binom{2}{1}=\frac{17\cdot 16\cdot 15\cdot 14\cdot 13\cdot 12}{6\cdot 5\cdot 4\cdot 3\cdot 2}\cdot \frac{11\cdot 10}{2}\cdot \frac{2}{1}=17\cdot 14\cdot 13\cdot 2\cdot 11\cdot 10\cdot 2=1361360$ .

However, we must account for symmetry in this problem. The $X$ has four possible rotations, thus the answer is

$\displaystyle \frac{1361360}{4}=340340$

I did the same thing but I focused on the nine first, then the two unique things. It leads to 17!/6!9!. From there, you divide by four for the answer. Anyways, that was a fun problem.

Guy Alves - 3 years, 11 months ago

Coconut Crossing

The answer is 340340.

1 solution

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