Suppose you\'re in charge of hanging 8 distinct paintings along a hallway: 2 pai
ID: 3123752 • Letter: S
Question
Suppose you're in charge of hanging 8 distinct paintings along a hallway: 2 paintings by Velazquez 2 paintings by Monet 1 painting by Picasso, 3 paintings by Kahlo (a.) How many different ways can you hang them up in a row? (b.) How many different ways can you hang them up in a row, if the Picasso painting is first? (c.) How many different ways can you hang them up in a row, so that either of the Monet paintings is first? (d.) How many different ways can you hang them up in a row, if you don't want the two paintings by Velazquez to be next to each other?Explanation / Answer
(a) Since there are 8 paintings, we can hang them in 8! = 40320 ways.
(b) Since Picasso painting is always first, we need to ignore it and find the number of ways to hang the remaining 7 paintings in a row.
This can be done in 7! ways = 5040 ways.
(c) The first painting is Monet's painting and can be chosen in 2 ways. Once fixed, we can ignore this and try to find the number of ways to hang the remaining 7 paintings which can be done in 7! ways
So this can be done in 2*7! = 10080 ways.
(d) First let us calculate the number of ways to hang the paintings such that Velzquez's paintings such that they are always next to each other. Since they are next to each other, let us count these two paintings as a 'big' painting. So we need to arrange this with the other six paintings, a total of seven 'small or big' paintings in all. This can be done in 7! ways.
The total number of ways to hang the paintings = 8!
The total number of ways to hang the paintings with the two Velazquez paintings = 7!
Therefore the total number of ways to hang the paintings such that the two Velazquez paintings are not next to each other = 8! - 7! = 40320 - 5040 = 35280
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