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.)
Choose the correct option:
A) 4
B) 5
C) 6
D) 12
E) 24
Let's start with 4 minimum number of colors   So single color code we can make 4 nos.   Now if we need to make 2 color combination out of 4 we can do so in 4!/2!*2! or 4*3/2 or 6   So total we can make 4+6= 
<gadget id="calculator">4+6</gadget>
<output>10</output>
10 color combinations but we have 12 boxes   So let's look at 5 we get 5 single color codes   and out 5 color choices, we can choose 2 in 5!/2!*3! ways or 10 ways.   So total we can have 5+10= 
<gadget id="calculator">5+10</gadget>
<output>15</output>
15 color combinations.   So, minimum number we need will be 5   Ans is 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 the colors in a pair does not matter.)