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NSWMaths Extension 1Quick questions

Combinatorics (ME-A1)

Quick questions on Combinations: counting unordered selections with nCr, the 2^n subset count, the symmetry nCr = nC(n-r), and restriction techniques (at least, complement, conditional, geometry)

8short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is every subset at once?
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Step back from a fixed size and count all subsets of an nn-element set together. Building a subset means going through the nn elements one at a time and deciding, independently, whether each is in or out, two choices per element. By the multiplication principle that is
What are restriction technique with combinations?
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The reshaping moves from the grouping, complement and cases page carry straight across to selections, because a combination is itself a count you can subtract from or split. Four moves cover almost every restricted-selection question.
What are geometry counts?
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A clean family of combination questions hides inside geometry. Given points with no three on a straight line, every unordered pair of points determines exactly one line (chord), and every unordered triple determines exactly one triangle, so the counts are simply combinations.
What is verifying a small case by listing?
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Whenever a count is small, the safest check is to list and tally. Take the triangles from 66 points {1,2,3,4,5,6}\{1,2,3,4,5,6\} on a circle. The formula gives 6C3=6×5×46=20^{6}C_{3} = \frac{6 \times 5 \times 4}{6} = 20. Listing every 33-element subset in increasing order gives
What is conditional: must contain a particular item?
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If a chosen object is forced in, put it in first, then choose the remaining places from the remaining objects. A team of 55 from 1111 that must include the captain CC has CC fixed, so the other 44 come from the remaining 1010: 10C4=210^{10}C_{4} = 210.
What is conditional: must exclude a particular item?
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If an object is banned, simply remove it and choose from what is left. A team of 55 from 1111 that must exclude an injured player XX chooses all 55 from the remaining 1010: 10C5=252^{10}C_{5} = 252. Combine the two: include CC and exclude XX leaves 44 to choose from 99, namely 9C4=126^{9}C_{4} = 126.
What are split a constrained selection into independent groups?
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When a committee must contain set numbers from disjoint pools, choose from each pool separately and multiply. A committee of 55 from 66 men and 88 women with exactly 22 men chooses 22 of the 66 men and 33 of the 88 women: 6C2×8C3=15×56=840^{6}C_{2} \times {}^{8}C_{3} = 15 \times 56 = 840.
What is at least / at most: count the complement?
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"At least kk" usually fragments into several size cases, so count the total and subtract the few unwanted cases. A committee of 44 from 55 women and 44 men with at least 22 women is the total 9C4=126^{9}C_{4} = 126 minus the committees with 00 or 11 woman, 4C4+5C1 4C3=1+20=21^{4}C_{4} + {}^{5}C_{1}\,^{4}C_{3} = 1 + 20 = 21, giving 126−21=105126 - 21 = 105.

All Maths Extension 1Q&A pages