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April 21st, 2016, 03:46 AM   #1
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An art gallery has a total of 11 paintings by a particular artist

An art gallery has a total of 11 paintings by a particular artist; of these, 5 are oil paintings and 6 are water colour paintings. The art gallery is asked to provide a special exhibition of this artist’s work but is restricted to showing only five of the paintings. Calculate the number ways in which the five paintings can be selected for the exhibition if :

there are no restrictions
one particular painting must be included
there must be exactly three watercolour paintings
there must be more watercolour paintings than oil paintings
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April 21st, 2016, 04:25 AM   #2
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1. I'm sure you can do this one yourself.
2. Choose four paintings out of the remaining 10.
3. Then there must be exactly two oil paintings. Choose two oil paintings out of five, then three water paintings out of 6.
4. Then any of these cases is possible:
5 watercolour, no oil
4 watercolour, 1 oil
3 watercolour, 2 oil
For each of these cases, calculate how many combinations are possible, then sum everything up.
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April 22nd, 2016, 08:18 AM   #3
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1) You can choose any one of the 11 paintings first, then any one of the remaining 10 second, then any one of the remaining 9, then any one of the remaining 8, then any one of the remaining 7. That makes a total of 11*10*9*8*7 which can also be written as 11!/6!.

2) Take out the given painting first. That leaves 10 painting from which to choose the remaining 4 paintings. There are, as above, 10*9*8*7= 10!/6! ways to do that.

3) There are 6*5*4 ways to select the three watercolors from the 6. There are 5*4 ways to choose two of the 5 oil paintings. There are 6*5*4*5*4 ways to do this.

4) "More watercolors than oil" could be:
5 watercolors, no oil pantings: 6*5*4*3*2
4 watercolors, one oil: (6*5*4*3)(5)
3 watercolors, two oils: (6*5*4)(5*4)

Add those.
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