Answer to Question #155123 in Combinatorics | Number Theory for maryam

Question

a flage with five stripes will be prepaerd for a school team .we have red ,blue,yellow ,green ,black and brown.strips cannot be left colorless and no more than one used in a strip. the same color cannot be in the tow adjacent stripes.when the flag is inverted ,the colors should appear in the same order.example;blue|yellow|black|yellow|blue….how many different flags can be obtained under these conditions?

Answer 1

First of all we note that the flag under these conditions is determined just by it's first three stripes (as the other two are determined by the condition of inversion of colors). Now knowing that there are six colors available, there is $6$ choices for the first stripe, $5$ choices for the second (as we can not have two stripes in a row with the same color, so we can't use the same as for the first stripe) and $5$ choices for the third (due to the same reasoning, we can't use the color used for the second stripe). Therefore there is a total of $6\times 5\times 5 = 150$ different flags satisfying the conditions.