# No. of ways to paint doors

• Dec 1st 2011, 05:31 AM
Punch
No. of ways to paint doors
A painter is given the job of painting the doors of 5 adjacent bedrooms using 5 colours: red, blue, yellow, green and orange.

If the owner of the house insists that the doors should be painted with at most two colours, find the number of ways which the painter could have the job done.
• Dec 1st 2011, 05:38 AM
Plato
Re: No. of ways to paint doors
Quote:

Originally Posted by Punch
A painter is given the job of painting the doors of 5 adjacent bedrooms using 5 colours: red, blue, yellow, green and orange.
If the owner of the house insists that the doors should be painted with at most two colours, find the number of ways which the painter could have the job done.

There are $\binom{2+5-1}{2}$ ways to paint each door. Why?
• Dec 1st 2011, 05:41 AM
Punch
Re: No. of ways to paint doors
Quote:

Originally Posted by Plato
There are $\binom{2+5-1}{2}$ ways to paint each door. Why?

• Dec 1st 2011, 07:14 AM
Plato
Re: No. of ways to paint doors
Quote:

Originally Posted by Punch

Wouldn't the answer be $(15)^5~?$
• Dec 1st 2011, 07:21 AM
Annatala
Re: No. of ways to paint doors
Punch, note that Plato said that's how many ways there are to paint each door.
• Dec 1st 2011, 04:51 PM
awkward
Re: No. of ways to paint doors
Quote:

Originally Posted by Punch
A painter is given the job of painting the doors of 5 adjacent bedrooms using 5 colours: red, blue, yellow, green and orange.

If the owner of the house insists that the doors should be painted with at most two colours, find the number of ways which the painter could have the job done.

I seem to have a different interpretation of this problem than the previous posters.

It seems to me that you would pick two colors and then paint each door a solid color, using one of the two colors previously selected. So

$\binom{5}{2} \times 2^5$

No multi-color doors for me, please!
• Dec 1st 2011, 05:18 PM
livinggourmand
Re: No. of ways to paint doors
(Thinking)All the doors can be painted with same color also....
it says "at most 2 color"
=no of ways of choosing 1 color *(5 doors) + no of ways of choosing 2 color *(5 doors)
(Thinking)
• Dec 1st 2011, 05:47 PM
Plato
Re: No. of ways to paint doors
Quote:

Originally Posted by awkward
I seem to have a different interpretation of this problem than the previous posters.

It seems to me that you would pick two colors and then paint each door a solid color, using one of the two colors previously selected. So
$\binom{5}{2} \times 2^5$
No multi-color doors for me, please!

Now that is a possible reading of that problem!
As many regular readers here may know, I in former life I was an editor for published test questions. As such, when I read a question here I naturally assume it has been fully vetted. However, in this case I think awkward has found what the true meaning of this question: doors have one color.
That should have been stated.
• Dec 1st 2011, 06:51 PM
Punch
Re: No. of ways to paint doors
Quote:

Originally Posted by awkward
I seem to have a different interpretation of this problem than the previous posters.

It seems to me that you would pick two colors and then paint each door a solid color, using one of the two colors previously selected. So

$\binom{5}{2} \times 2^5$

No multi-color doors for me, please!

That gives 320 and adding the ways of painting all doors in one colour gives 325 ways. But the correct answer is 305.

I can't seem to spot a correct answer in all the posts.

And yes, do assume that a door should be painted with only one colour.
• Dec 2nd 2011, 05:16 AM
livinggourmand
Re: No. of ways to paint doors
Quote:

Originally Posted by Punch
That gives 320 and adding the ways of painting all doors in one colour gives 325 ways. But the correct answer is 305.

I can't seem to spot a correct answer in all the posts.

And yes, do assume that a door should be painted with only one colour.

Including akwards post

Painting with 2 colors

total no. painting 5 doors with 2 colors = $2^5$
now this combination will also be having case where all the doors were painted same with color...
no of such cases = 2
total no. painting 5 doors with 2 colors(choosing 2 colors of 5) = $\binom{5}{2}(2^5-2)=300$

Painting with 1 colours

$\binom{5}{1} =5$

so total = 305