A certain gallery is hanging a series

VJesus12 wrote:A certain gallery is hanging a series of 7 paintings. All the paintings will be exhibited in a row along a single wall. Exactly 2 of the paintings are on panel, the remainder are on canvas. In how many ways can the paintings be exhibited if the works on panel must be the second and sixth in the row?

A. 240
B. 200
C. 122
D. 80
E. 16

The OA is A.

What is the formula that I should use to get the answer? I tried to solve it but I couldn't. Experts, I need your help. Thanks.

There are 2! ways to arrange the 2 paintings on panel and 5! ways to arrange the 5 paintings on canvas. Therefore, the total number of ways to arrange the 7 paintings is 2! x 5! = 2 x 120 = 240.

Answer: A

VJesus12 wrote:A certain gallery is hanging a series of 7 paintings. All the paintings will be exhibited in a row along a single wall. Exactly 2 of the paintings are on panel, the remainder are on canvas. In how many ways can the paintings be exhibited if the works on panel must be the second and sixth in the row?

A. 240
B. 200
C. 122
D. 80
E. 16

Start with the most restrictive condition

Place one panel painting.
This must be in 2nd or 6th position.
So, we can complete this stage in 2 ways

Place the remaining panel painting.
Since we already placed one panel painting, there's only 1 way to place the next painting.

From here, we have 5 paintings and 5 positions remaining
So, we can place the first remaining painting in 5 positions.
We can place next remaining painting in 4 positions.
We can place next remaining painting in 3 positions.
We can place next remaining painting in 2 positions.
We can place next remaining painting in 1 position.

Total arrangements = (2)(1)(5)(4)(3)(2)(1) = 240

Answer: A