# Problem of the Week Problem B and Solution Three-Chip Stacks

## Problem

Chipâ€™s favourite snack food is Dingles potato chips. He enjoys layering 3 chips together to make a stack of interesting flavours. If he only has Regular flavoured Dingles, there is only one way to stack the three chips. That is Regular, Regular and Regular which can be written as rrr. The order in which the chips are layered does not matter. For example, if we have Regular and BBQ Dingles then the stack BBQ, BBQ and Regular (bbr) is the same as BBQ, Regular and BBQ (brb).

1. If he has Regular and BBQ flavoured Dingles, list all the different stacks of 3 chips he can make.

2. If he has three flavours of chips, Regular, BBQ, and Vinegar, list all the different stacks of 3 chips he can make.

3. If he adds a fourth flavour, Ketchup, list all the different stacks of 3 chips he can make.

Suggestion: You may use the following boxes to help you organize the possible stacks. You may also want to use r to represent Regular, b for BBQ, v for Vinegar, and k for Ketchup.

## Solution

The lists for parts a), b), and c) are given in the tables below. A solution using an exhaustive process is shown after these lists.

List for a):

 rrr rrb rbb bbb

List for b):

 rrr rrb rrv rbb rvv rbv bbb bbv bvv vvv

List for c):

 rrr rrb rrv rrk rbb rvv rkk rbv rbk rvk bbb bbv bbk bvv bkk bvk vvv vvk vkk kkk

Solution for a):

Number of
Regular
Number of
BBQ
Possible
Stack
3 0 rrr
2 1 rrb
1 2 rbb
0 3 bbb

Solution for b):

Number of
Regular
Number of
BBQ
Number of
Vinegar
Possible
Stack
3 0 0 rrr
2 1 0 rrb
2 0 1 rrv
1 2 0 rbb
1 0 2 rvv
1 1 1 rbv
0 3 0 bbb
0 2 1 bbv
0 1 2 bvv
0 0 3 vvv

Solution for c):

Number of
Regular
Number of
BBQ
Number of
Vinegar
Number of
Ketchup
Possible
Stack
3 0 0 0 rrr
2 1 0 0 rrb
2 0 1 0 rrv
2 0 0 1 rrk
1 2 0 0 rbb
1 0 2 0 rvv
1 0 0 2 rkk
1 1 1 0 rbv
1 1 0 1 rbk
1 0 1 1 rvk
0 3 0 0 bbb
0 2 1 0 bbv
0 2 0 1 bbk
0 1 2 0 bvv
0 1 0 2 bkk
0 1 1 1 bvk
0 0 3 0 vvv
0 0 2 1 vvk
0 0 1 2 vkk
0 0 0 3 kkk