probability

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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 fo one or two colors, what is the minimum number of colors needed for the coding

5

[mental]

Let n be the number of colours

total choices possible = n + nC2
Requirement n + nC2 = atleast 12

if n=3:
3 + 3C2 = 6 INSUFFICIENT

if n=4:
4 + 4C2 = 10 INSUFFICIENT

if n=5:
5 + 5C2 = 15 more than 12

we see that 4 colours wil result in only 10 possibilities.

So min number of colours required: 5

[vittalgmat]

This can be solved another way.

So here it goes.
with 1 color: u can represent 2 choices.
color present = choice 1;
color absent = choice 2

extending this
2 colors --> 2*2 or 2^2 = 4 choices.

3 colors -> 2*2*2 or 2^3 = 8 choices

4 colors -> 2*2*2*2 or 2^4 = 16 choices.

So 4 colors is sufficient to represent 12 choices.

my guess is this is a moderate tough problem in the 600 -650 range.. But that is my guess.

As an aside, in the world of computerscience/digital electronics, this logic is very similar to finding out how many bits (binary digits) are needed to represent 'n' number of different characters.

HT Helps