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 colors in a pair does not matter)
Options:
A) 4  B) 5  C) 6  D) 12  E) 24
Let # of colors needed be nn, then it must be true that n+C2n≥12n+Cn2 - # of ways to choose the pair of different colors from nn colors when order doesn't matter) --&gt; n+n(n−1)/2≥12 --&gt; 2n+n(n−1)≥24 --&gt; n is an integer (it represents # of colors) n≥5--&gt; nmin=5   Answer: 5.
<result>B</result>
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 colors in a pair does not matter)