Counting, permutations, combinations and sets

01 / UNDERSTAND & APPLY

Define one outcome precisely before counting

A count is meaningful only after defining when two outcomes are different. If the outcome is a committee, rearranging the same members does not create a new committee. If it is president and treasurer, swapping the two people creates a different assignment. Order means distinct roles or positions, not merely the order in which you happen to make choices.

Separate restrictions such as repetition allowed, items distinct, or at least one category required. A formula selected before these decisions can produce a polished answer to the wrong sample space.

02 / UNDERSTAND & APPLY

Multiply sequential choices; add disjoint alternatives

If a task consists of stages with a fixed number of options at each stage, multiply their counts. If options available later depend on earlier choices, use correct conditional branch counts. Add counts for alternatives only when their outcome sets do not overlap.

For three shirts and four trousers, 3·4=12 outfits if any shirt can pair with any trousers. If a particular pairing is forbidden, subtract it or count allowed branches; do not continue using the unrestricted product.

03 / UNDERSTAND & APPLY

Distinct ordered selections use falling products

Arranging r distinct objects chosen from n distinct objects without repetition gives n(n−1)…(n−r+1)=n!/(n−r)!. The first position has n choices and each selection removes one. For all n objects, the count is n!. By definition 0!=1, consistent with one empty arrangement.

If repetition is allowed across r positions, the count is nʳ instead. Factorials apply to selections without replacement, not automatically to every ordering problem.

04 / UNDERSTAND & APPLY

Remove the internal order when only membership matters

Selecting r members from n without order gives C(n,r)=n!/[r!(n−r)!]. The permutation count overcounts each chosen group r! times, once for each arrangement. Dividing by r! removes that duplication.

C(n,r)=C(n,n−r), since selecting who is included also selects who is excluded. This symmetry can simplify calculations. Do not divide by r! when roles or order actually distinguish outcomes.

05 / UNDERSTAND & APPLY

Build restrictions into stages or use a complement

For a committee with a specified composition, choose within each category and multiply: choosing two from one group and one from another gives C(a,2)C(b,1), assuming the groups are disjoint. For at least one from a category, count all groups and subtract those with none.

For arrangements with identical repeated objects, divide n! by the factorial of each repeated multiplicity, because swaps among indistinguishable copies do not produce new arrangements. A problem about actual labeled objects may not permit that division.

06 / UNDERSTAND & APPLY

Fill disjoint regions before adding totals

For two sets, divide the population into A only, both, B only and neither. These four regions are disjoint. Set totals include the overlap, so A only=|A|−|A∩B|. The union counts everyone in at least one set: |A∪B|=|A|+|B|−|A∩B|.

If the total population is N, neither=N−|A∪B|. “Exactly one” is A only plus B only, not the union. A Venn circle’s area is usually schematic; read labeled counts rather than estimating proportions from its drawn size.

07 / UNDERSTAND & APPLY

Three-set inclusion-exclusion corrects the corrections

For three sets, the union equals the sum of the three set sizes minus all three pairwise intersections plus the triple intersection. Pairwise intersection counts include the triple region unless explicitly described as “only.” The triple region was added three times then subtracted three times, so it must be added once.

A region-first diagram is often safer than memorizing a formula. Place the triple count first, subtract it from each inclusive pairwise count to obtain pair-only regions, then fill the single-only regions. Negative resulting counts signal inconsistent data or a mistaken interpretation.

08 / UNDERSTAND & APPLY

Match numerator and denominator to the same outcome type

Counting can calculate probability when all counted outcomes are equally likely. If the denominator counts unordered committees, the numerator must count unordered committees too. Mixing an ordered numerator with an unordered denominator exaggerates the probability.

An original random-selection protocol must justify uniform selection. Choosing a category uniformly and then a person uniformly within it does not select every person equally when category sizes differ.

09 / UNDERSTAND & APPLY

Choose a method and answer the requested question

All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.

A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.

A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.

10 / UNDERSTAND & APPLY

Original practice: order, repetition and overlap

Decide the counting model before computing.