0
$\begingroup$

If we have a uniform rod of length 1m, which is to be decorated by dividing the surafce into five 20cms bands, and colour each band red or blue, how many distinct ways are there to achieve this?

How to account for the 'distinct' part of the question?

$\endgroup$
3
  • $\begingroup$ "How to account for" is the second question. The first question is, "how is distinct defined" in the first place. If the rod is colored red at one end (one band) and all the other bands are blue, is coloring the "left" end red considered the same as coloring the "right" end red, or are they considered "distinct"? $\endgroup$
    – user325968
    Nov 25, 2017 at 15:12
  • $\begingroup$ They are considered the same @mathguy $\endgroup$
    – kauray
    Nov 25, 2017 at 15:18
  • 1
    $\begingroup$ Good - that should be included in the problem statement. (How did you know the answer? That suggests it WAS in the problem statement, otherwise how would YOU know that, without asking the teacher?) $\endgroup$
    – user325968
    Nov 25, 2017 at 15:19

1 Answer 1

1
$\begingroup$

For simplicity, let's initially "fix" the rod (it can't be rotated 180 degrees) and label the bands a, b, c, d, e.

There are $2^5 = 32$ ways to color the five bands with two colors. Now we must count how many of them are "distinct".

Some of the colorings are symmetric - if you rotate the rod 180 degrees, the new coloring will coincide with the initial one. To completely determine a symmetric coloring, you need to know the coloring of a, b, c; so there are 8 symmetric colorings.

The remaining 24 colorings fall into pairs - if you "rotate the rod 180 degrees" starting with one of these colorings, you get a different coloring. So, out of these 24 colorings, only 12 are considered "distinct" under the definition of "distinct" as clarified in the OP's comment.

Adding back the symmetric colorings, there are 20 distinct colorings overall.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .