'manpreet singh wrote:How many different ways can 3 cubes be painted if each cube is painted one color and only the 3
colors red, blue, and green are available? (Order is not considered, for example, green, green, blue is considered the same as green, blue, green.)
(A) 2
(B) 3
(C) 9
(D) 10
(E) 27
Here's a "mathematical" approach:
WARNING: I'll begin with a somewhat convoluted setup, but it should all make sense shortly.
Start with 5 circles: OOOOO
Choose any 2 circles, and replace them with a line.
Some examples: O|O|O or O||OO
Notice that the 2 lines divide the circles into 3 separate regions.
We'll say that:
- the number of circles in the left-most region represents the number of red cubes
- the number of circles in the middle region represents the number of blue cubes
- the number of circles in the right-most region represents the number of green cubes
So, we can take O|O|O and think of it as
O|
O|
O. In other words, O|O|O represents 1 red cube, 1 blue cube and 1 green cube.
Similarly, we can take O||OO and think of it as
O||
OO. In other words, O||OO represents 1 red cube, 0 blue cubes and 2 green cubes.
We can take OO|O| and think of it as
OO|
O||. In other words, OO|O| represents 2 red cubes, 1 blue cube and 0 green cubes.
We can take ||OOO and think of it as ||
OOO. In other words, ||OOO represents 0 red cubes, 0 blue cubes and 3 green cubes.
So, each time we select 2 circles and replace them with lines, we get a different possibility for the cube colors.
So, the question boils down to,
"In how many ways can we select 2 of the 5 circles?"
Since the order in which we select the circles does not matter, we can use combinations.
We can select 2 circles from 5 circles in 5C2 ways ([spoiler]= 10 ways = D[/spoiler])
If anyone is interested, we have a free video on calculating combinations (like 5C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789
Cheers,
Brent
