There are 6 landscapes, 5 portraits, and 7 still lifes available for an art display. Two of each type of painting are selected. The paintings are each hung in one of six locations in the gallery. In how many ways could the art be displayed?

  • $\begingroup$ What are your thoughts? $\endgroup$ – Jared Oct 6 '16 at 2:11
  • $\begingroup$ Are the paintings unique? ie. Is landscape and landscape 2 considered different from one another? $\endgroup$ – suomynonA Oct 6 '16 at 2:15

I'm assuming all the landscapes, portraits, and still lifes are unique within their own categories, correct me if I am wrong.

$6 \choose 2$ ways to choose two paintings from landscapes.

$5 \choose 2$ ways to choose two paintings from portraits.

$7 \choose 2$ ways to choose two paintings from still lifes.

Multiply these together, and you get $${6\choose2} \cdot {5 \choose 2} \cdot {7 \choose 2}=15 \cdot 10 \cdot 21=3150$$There are $6$ ways to place the first painting, $5$ to place the second, etc. So the number of ways to place the paintings is $$6!=720$$Multiply the two numbers together, and you get $$3150 \cdot 720=2,268,000$$

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