A necklace is made by stringing N individual beads together in the repeating pattern red bead, green bead, white bead, blue bead and yellow bead. If the necklace design begins with a red bead and ends with a white beac, then N could euqal

A. 16
B. 32
C. 41
D. 54
E. 68

According to the answer, the number of beads in this design is 3 more than some multiple of 5. This can be expressed as 5n+3, where n is an integer. I don't understand this explanation. Thank you.

RGWBY is the bead pattern and it repeats. So, the 3rd, 8th, 13th, 18th... beads will be W.
(This is similar to an AP series). So, number of beads - 3 has to be divisible by 5 for it to be a White bead. 68 - 3 = 65 = 5*13. Answer is E.

givemeanid, thanks for the explanation!
-dan

Thank you, givemeanid.

greenpepper

i recognize that this is a combinatorics problem but it is still unclear. Can i get a fuller explanation on the approach? I don't understand how you get 68? :?

It's actually a sequences question - you want to use the pattern to get to the answer.

The pattern is RGWBY RGWBY RGWBY etc (keeps repeating)
So, for example, the red beads are #1, #(1+5), #(1+5+5), etc, or 1, 6, 11, etc.
The green beads are #2, #(2+5), #(2+5+5), etc, or 2, 6, 12, etc
and so on for the other colors

We want to find what N could be, and N represents the last bead. We're told the last bead is white, so what are the options for white? #3, #(3+5), #(3+5+5), etc, or 3, 8, 13, etc.

I can figure out what works from my answer choices in two different ways.

The easiest way (unique to this problem set-up): each time, I'm adding 5. So, for N, it could be 3, 8, 13, 18, 23, 28... noticing a pattern? It always ends with either 3 or 8. The only answer choice that ends with either 3 or 8 is E.

The common way (can use on any problem of this type) is to set it up algebraically. For white, I start with 3 and then each time I add 5 more. That can be expressed as 3 + 5x. The first white bead is just at 3, the second white bead is at 3 + 5(1) = 8, the third white bead is at 3 + 5(2) = 13, and so on. This means the answer will be 3 + some multiple of 5. Subtract 3 from every answer choice. The right answer should be a multiple of 5 (because we've removed the first 3). Only E fits this requirement.