# In how many ways can Ben paint his apartment?

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In summary, Ben has 14 walls in his apartment and has enough paint to cover 10 of them with one color and the remaining 4 with another color. By using combinations, there are 1,001 ways for Ben to paint his apartment, as he can either choose the 10 walls for one color or the 4 walls for the other and they are equivalent.
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The rooms of Ben’s apartment has 14 walls. He has enough paint to cover 10 of these walls with one color and the rest with another color. In how many ways could Ben paint his apartment ?

This problem I used combinations for 10 walls for one color and 4 walls for one color; Here my calculations:

14 walls in total, 10 walls in one colour and the rest (4 walls) in another colour.

For 10 walls: nCr = 14C10 = n!/(n-r)!r! = (14)!/(14-10)!(10)! = (14)!/(4)!(10)!

=(14x13x12x11x10!)/(4x3x2x1)(10)! = (24,024)/(24) = 6,006

For 4 walls: nCr = 14C4 = n!/(n-r)!r! = (14)!/(14-4)!(4)! = (14)!/(10)!(4)!

=(14x13x12x11x10!)/(4x3x2x1)(10)! = (24,024)/(24) = 6,006

i wonder if I am in the right direction ? Pls and Thank you

I think you are definitely on the right track. We could either look at the number of ways to choose 10 from 14 or the number of ways to choose 4 from 14, as after all these will necessarily return the same number since:

$$\displaystyle {n \choose r}={n \choose n-r}$$

$$\displaystyle \frac{n!}{r!(n-r)!}=\frac{n!}{(n-r)!(n-(n-r))!}$$

$$\displaystyle \frac{n!}{r!(n-r)!}=\frac{n!}{r!(n-r)!}\quad\checkmark$$

$$\displaystyle {14 \choose 4}=\frac{14!}{4!(14-4)!}=\frac{14!}{4!10!}=\frac{14\cdot13\cdot12\cdot11}{4!}=7\cdot13\cdot11=1001$$

Your work was correct up until the final division where you made a slight error.

Thank you, guess did most of the work and didnt finish it . So do I have to multiply 14C10 x 14C4 to get the final answer ?

No, the final answer is that there are 1,001 ways for Ben to paint his apartment. He can either choose the 10 walls for one color, or the 4 walls for the other. In either case he will find there are 1001 ways to do so, and that they are equivalent.

Thank you very much

## 1. How many different colors can Ben use to paint his apartment?

The number of colors Ben can use depends on his personal preference and the availability of paint options in his area. However, in theory, there are infinite colors that Ben can use to paint his apartment.

## 2. Can Ben use the same color for different rooms in his apartment?

Yes, Ben can use the same color for different rooms in his apartment. This would be considered one way of painting his apartment.

## 3. Does the type of paint affect the number of ways Ben can paint his apartment?

Yes, the type of paint can affect the number of ways Ben can paint his apartment. For example, if Ben chooses to use two different types of paint (e.g. matte and glossy), this would be considered two different ways of painting his apartment.

## 4. Can Ben mix and match different paint colors to create new colors?

Yes, Ben can mix and match different paint colors to create new colors. This would increase the number of ways he can paint his apartment.

## 5. Is there a limit to the number of ways Ben can paint his apartment?

Technically, there is no limit to the number of ways Ben can paint his apartment. However, in practical terms, there may be limitations such as budget, time, and availability of resources that may affect the number of ways he can paint his apartment.

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