DP by final choice
Count Ordered Token Programs
01. The question
A program consists of positive-valued tokens. Count ordered token sequences whose values total the requested amount; each token value can be used repeatedly. Original program scenario; [1,3] and [3,1] are different programs.
Before revealing the idea, predict the result and describe what information you would need to track.
02. Worked example
Input[1, 3, 4]; amount = 5
Output6
- The empty program is the only program totaling 0.
- Programs totaling 4 can end with 1, 3, or 4: 2 + 1 + 1 = 4 ways.
- At total 5 the final tokens contribute 4 + 1 + 1 = 6 ways.
Try your code
Define solve(data, parameter="") and return your result. Your code runs locally, not on a server.
Edit your test cases
Use 1-10 objects with data and an optional string parameter. Expected outputs are computed from the lesson. An optional expected field must be correct.
Input contract & reference code
Example data: [1,3,4]. Parameter: "5". Numeric parameters arrive as strings; convert them with int(parameter) when needed.
The example and generated cases use the question’s exact variant. This checks behavior on a small test batch—not performance on every possible input.
Python downloads only when you run code, using compressed runtime assets. Code is not saved or uploaded. Enable JavaScript to run it.
Printed output
Five-second execution limit per batch. Standard-library Python only; additional packages and network access are unavailable. Stop ends execution without blocking the lesson.
03. The idea
Try sketching your own approach first. What would you track, and why would it work?
Reveal the idea
- Set ways[0] = 1.
- Process totals outside the token loop so order matters.
- For each possible last token add the ways to total minus that token.
04. Watch it happen
Use Step to move slowly, or Play to follow the whole trace. Change a preset or load your own input.
Input & settings
Red = current item / blue = active window or cells / green = completed work. A value docks at its actual destination; the original input stays unchanged. Positions start at 0.
Working state / what changes
Ready to begin
Enable JavaScript to step through the animation. The worked example and solution below remain readable without it.
Python / why it changes
The line behind the step
def solve(data, parameter=''):
target = int(parameter)
ways = [0] * (target + 1)
ways[0] = 1
for total in range(1, target + 1):
for choice in data:
if choice <= total:
ways[total] += ways[total - choice]
return ways[target]How to use this walkthrough
Hover or focus a control for its label. Play forms the input and follows values into the working state. Step shows one change; Back and the slider restore exact snapshots. Reduced motion uses instant updates. Saved snapshots and sorted heap displays are teaching overhead, not algorithm complexity.
05. Python solution
def solve(data, parameter=''):
target = int(parameter)
ways = [0] * (target + 1)
ways[0] = 1
for total in range(1, target + 1):
for choice in data:
if choice <= total:
ways[total] += ways[total - choice]
return ways[target]06. Cost & edge cases
- Time
- O(target × number of token values)
- Space
- O(target)
- Target 0 returns 1 even with no tokens.
- An unreachable positive target returns 0.
- Use 0-8 distinct values from 1 to 8 and a target from 0 to 16; zero tokens are rejected to prevent infinitely many programs.