A company that ships boxes to a total of 12 distribution centers …Why B is the answer ?
A company that ships boxes to a total of 12 distribution centers uses color coding to identify each center. If either a single color or a pair of two different colors is chosen to represent each center and if each center is uniquely represented by that choice of one or two colors, what is the minimum number of colors needed for the coding? (Assume that the order of the colors in a pair does not matter.)
a. 4
b. 5
c. 6
d. 12
e. 24
Hi Prem,
There are two ways to approach this question.
- Using the answer choices
- Using the concept of combinations
Using the answer choices let us start with the answer option C – 6
Let the 6 colors be Y, B, G, R, O, P. Since there are 12 distribution centers, let us check if 6 is the minimum number of colors to name the 12 distribution centers.
- Y
- B
- G
- R
- O
- P
- YB
- YG
- YR
- YO
- YP
- BG
- BR ……..
So clearly 6 is not the minimum number. We can eliminate 6, 12 and 24.
If we have 5 colors say Y, B, G, R, O then
- Y
- B
- G
- R
- 0
- YB
- YG
- YR
- YO
- BG
- BR
- BO…… a lot more combinations are possible. So the only possible answer is 4.
The shorter way to work this question out is to use a combination (the question states that the order of the colors so not matter). If we have 4 colors, then there are 4C1 = 4 ways of naming a distribution center with one color, 4C2 = 6 ways of naming a distribution center with 2 colors, 4C3 = 4 ways of naming a distribution center with 3 colors and 4C4 = 1 way of naming a distribution center with 4 colors. So 4 + 6 + 4 + 1 = 15. So we can name 15 distribution centers with 4 colors.
Hope this helps!
