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NSWMaths Extension 1Quick questions

Combinatorics (ME-A1)

Quick questions on Combinations: counting unordered selections with (nr)\binom{n}{r}

15short 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 the combination formula?
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The number of ways to choose rr unordered objects from nn distinct objects is
What is key identities?
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$(nr)=(nn−r).\binom{n}{r} = \binom{n}{n - r}.$
What is complementary counting?
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For "at least kk" or "at most kk" problems, it is often easier to count the complement.
What is splitting into groups?
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To split nn objects into groups of sizes r1,r2,…,rkr_1, r_2, \dots, r_k (with r1+⋯+rk=nr_1 + \dots + r_k = n):
What is counting with constraints?
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"Includes object A": A is in the subset. Choose the remaining r−1r - 1 from the other n−1n - 1: (n−1r−1)\binom{n - 1}{r - 1}.
What is symmetry?
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(2018)=(202)=20⋅192=190\binom{20}{18} = \binom{20}{2} = \frac{20 \cdot 19}{2} = 190. Faster than computing (2018)\binom{20}{18} directly.
What is two groups?
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A team of 77 has 44 defenders and 33 attackers. Select from 99 defenders and 66 attackers.
What is at least?
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How many three-letter subsets of {\{A, B, C, D, E, F}\} include at least one vowel?
What is includes a specific element?
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A book club selects 44 books from 1515. How many selections include "War and Peace"?
What is splitting into three groups?
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Split 99 people into three teams of 33.
What is pascal in action?
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Verify (63)=(52)+(53)\binom{6}{3} = \binom{5}{2} + \binom{5}{3}.
What is "Includes object A"?
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A is in the subset. Choose the remaining r−1r - 1 from the other n−1n - 1: (n−1r−1)\binom{n - 1}{r - 1}.
What is "Excludes object A"?
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A is not in the subset. Choose all rr from the other n−1n - 1: (n−1r)\binom{n - 1}{r}.
What is "Includes A or B but not both"?
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count includes-A-excludes-B plus includes-B-excludes-A.
What is "At least one of A, B, C"?
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use complementary counting or inclusion-exclusion.

All Maths Extension 1Q&A pages