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NSWMaths Extension 1Quick questions

Combinatorics (ME-A1)

Quick questions on Permutations: counting ordered arrangements with the multiplication principle

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 multiplication principle?
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If a procedure can be performed in n1n_1 ways at step 1, and (independent of step 1) in n2n_2 ways at step 2, …\dots, and nkn_k ways at step kk, then the total number of ways to complete the procedure is n1⋅n2⋅⋯⋅nkn_1 \cdot n_2 \cdot \dots \cdot n_k.
What is permutations of nn distinct objects?
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The number of ways to arrange all nn distinct objects in a row is
What is permutations of rr from nn?
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The number of ways to choose and arrange rr objects from nn distinct objects is
What is permutations with repeats?
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If you have nn objects of which n1n_1 are alike, n2n_2 are alike, …\dots, nkn_k are alike (with n1+n2+⋯+nk=nn_1 + n_2 + \dots + n_k = n), the number of distinct arrangements is
What is circular permutations?
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The number of distinct circular arrangements of nn distinct objects is (n−1)!(n - 1)!. Reasoning: fix one object to break the rotational symmetry, then arrange the remaining n−1n - 1 linearly.
What is restrictions?
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Two objects must be together: glue them together as a single block, arrange as if n−1n - 1 objects, then multiply by 2!2! for the internal arrangement of the block.
What is summary recipe?
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1. Identify whether order matters (yes for permutations, no for combinations). 2. Identify whether repetition is allowed (with-repeat formulas are nrn^r).
What is rr from nn?
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How many four-digit codes using digits 11 to 99 with no repeated digit? 9P4=9⋅8⋅7⋅6=3024{}^9 P_4 = 9 \cdot 8 \cdot 7 \cdot 6 = 3024.
What is word with repeats?
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How many arrangements of the letters in MISSISSIPPI? Letters: M, I (4), S (4), P (2). Total 1111 letters.
What is circular?
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In how many ways can 77 people sit at a round table? (7−1)!=720(7 - 1)! = 720.
What is together restriction?
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How many arrangements of 55 people in a row if Alice and Bob must sit together?
What is not together restriction?
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Same setup but Alice and Bob must not sit together.
What is fixed position?
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How many arrangements of the letters MATHS such that the M is in the first position? Lock the M: arrange the remaining 44 letters in 4!=244! = 24 ways. :::
What is two objects must be together?
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glue them together as a single block, arrange as if n−1n - 1 objects, then multiply by 2!2! for the internal arrangement of the block.
What is two objects must not be together?
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count total arrangements minus the "together" count.

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