Separate disjoint alternatives from sequential choices and use complements.
Loading your saved responses…
Try
Choosing one of three sandwiches and one of two drinks gives six meals if every pairing is allowed. Choosing either a sandwich alone or a drink alone instead gives five possibilities. The word “or” does not automatically justify addition: the alternatives must be disjoint.
Predict
Predict what an outcome actually records. The sum rule counts disjoint alternatives. The product rule counts successive choices with a fixed number of available options at each stage. If branch sizes differ, add their separate counts rather than pretending every branch has the largest size.
Experiment
Draw a two-stage tree with first choice A or B. Give A three endings and B two endings. Label all five leaves. Change the rule to allow the third ending after B, and identify the single new outcome responsible for the change from five to six.
Five valid code outcomes: A has endings 0,1,2 while B has only 0,1. The schematic tree has five leaves; there is no B2 branch.
Understand
Worked complement: binary strings of length four number 2⁴=16. Exactly one has no 1, so 15 contain at least one 1. To require both symbols, exclude all zeros and all ones, giving 14. The subtraction is justified because the forbidden strings are distinct at positive length.
A counting solution needs a correspondence between counted choices and intended outcomes. If two paths in a decision tree produce the same final object, counting leaves overcounts. If a valid object has no path, the construction undercounts.
Invalid rule: multiply 5×5 when selecting two distinct people without replacement. After the first selection, only four remain. Conversely, forbidding reuse when the task allows repeated symbols incorrectly reduces the count.
Apply
In each practice answer, state whether order matters, whether repetition is permitted and whether all pairings are allowed. For a complement, name the full universe and the exact excluded subset rather than subtracting a vaguely described collection.
Continue
Depth: a variable-branch product can be expressed as a sum over first choices. Probability uses the same partition logic, but a count becomes a probability only under an explicit equally likely outcome model. This lesson counts possibilities without assuming a random mechanism.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
A meal chooses one of 4 mains and one of 3 drinks, with every pairing allowed. How many meals?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
How many binary strings of length 6 contain at least one 1?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
When may |A∪B| be calculated as |A|+|B| without correction?
The product 5×4 counts two selections from five objects when:
A code starts with A or B. A permits endings 0,1,2; B permits endings 0,1. Count and explain why 2×3 is wrong.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove that binary strings of length n containing both symbols number 2ⁿ−2 for every integer n≥1; explain the n=0 exception.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
