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 ?
If he only has enough to paint 14 walls, then there are 14C10 ways to pick the 10 walls for color #1.
If he enough of the 2nd color to paint any or all of the walls, then
color1 + color2
10+4: 14C10
9+5: 14C9
...
1+13: 14C1
0+14: 1
Now add them all up
14C10 for the first 10 - first color
14C4 for rest of the walls (4) second color
14C10+14C4
That was my understanding ...
10+4: 14C10
9+5: 14C9
...
1+13: 14C1
0+14: 1
could not follow .... Thx
since you only have two colors, the 14C10 ways to choose the 1st color walls leaves you no choice for the 2nd color. You always have to pick all 4 of the unchosen walls, right?
Thank you Pal,
Kind of confused . If you can be more specific I appreciate.
To solve this problem, we need to determine the number of ways Ben can choose the 10 walls to paint in one color out of the 14 available walls in his apartment.
This problem can be solved using a combination formula, where the order of selection does not matter. We can use the combination formula nCr, where n represents the total number of walls and r represents the number of walls to be painted in one color.
In this case, we have 14 walls and want to choose 10 walls to be painted in one color. So the formula becomes:
C(14, 10) = 14! / (10! * (14 - 10)!)
Let's break down the steps to solve this:
1. Calculate the factorial of 14: 14! = 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 87,178,291,200.
2. Calculate the factorial of 10: 10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3,628,800.
3. Calculate the factorial of (14 - 10): (14 - 10)! = 4 * 3 * 2 * 1 = 24.
Now let's substitute these values back into the combination formula:
C(14, 10) = 87,178,291,200 / (3,628,800 * 24)
= 87,178,291,200 / 87,139,200
= 1,000.
Therefore, there are 1,000 ways Ben can paint his apartment by choosing 10 walls to be painted in one color and the rest in another color.