The Gagosian art gallery in New York City has 20 stored paintings but has just m

Question

The Gagosian art gallery in New York City has 20 stored paintings but has just made room to display several of them. Seven paintings will be randomly selected and offered to the public for sale.

a.) How many different collections of the 7 paintings are possible?

b.) Suppose 10 of the 20 stored works are by the same local artist. What is the probability that all 7 of the selected paintings will be by this artist?

c.) the featuere room in the gallery recieves the most attention, and the order in which the paintings are displayed in this room is related to buyer interest. Suppose the 7 selected paintings will be placed in this featured room. How many different arrangements are possible?

Explanation / Answer

( a )

This is the case of combination.

Here we have n = 20 and r = 7

nCr = n! / (r! * (n-r)!)

20C7 = 20! / ( 7! * (20-7)!)

         = 20! / ( 7! *13!)

        = (20 * 19 * 18 * 17 * 16 * 15 *14 * 13! ) / ( 7! *13!)

        = 77520

Answer: 77520

( b )

Probability = (10C7) / (20C7)

10C7   = 10! / ( 7! * (10-7)! ) = 10! / ( 7! * 3! ) = 120

20C7= 20! / ( 7! * (20-7)!) = 77520

Probability = 120 / 77520

                    = 0.002

Answer: 0.002

( c )

This is the case of permutation.

nPr = n! / (n-r)!

N = 7 and r = 7

7P7 = 7! / ( 7 – 7 ) ! = 7! / 0! = 7! / 1 = 7! = 5040

Answer: 5040

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