Test 2 Prep
MAT 200
Integer Properties
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Know how to compute div and mod, which are respectively d and r in the following:
𝑎 = 𝑑𝑞 + 𝑟, 0 ≤ 𝑟 < 𝑑
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Note the condition on r!
lcd(m,n) x gcd(m,n) = mn
Mathematical Induction
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Establish the base case
State the induction hypothesis, 𝑃(𝑘)
Show that 𝑃(𝑘) ⇒ 𝑃(𝑘 + 1).
Counting
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Product rule
Sum rule
|𝐴 ∪ 𝐵 | = |𝐴| + |𝐵 | − |𝐴 ∩ 𝐵|
𝑚
Combinations without replacement: ( ) = 𝐶(𝑚, 𝑛)
𝑛
Combinations with replacement: 𝐶(𝑛 + 𝑟 − 1, 𝑟)
Permutations without replacement: 𝑃(𝑚, 𝑛)
Permutations with replacement: 𝑛𝑟
𝑛!
Distinct arrangements of letters with repeated letters: 𝑛 !𝑛 !⋯𝑛 !
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Binomial theorem expansions: (𝑥 + 𝑦)𝑛 = ∑𝑛𝑗=0 𝐶(𝑛, 𝑗)𝑥 𝑛−𝑗 𝑦 𝑗
𝑘
Probability
|𝐸|
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𝑃(𝐸) =
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Conditional probability: 𝑃(𝐴 |𝐵) =
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Bernoulli trials: probability of k successes is 𝐶(𝑛, 𝑘)𝑝𝑘 (1 − 𝑝)𝑛−𝑘
|𝑆|
𝑃(𝐴∩𝐵)
𝑃(𝐵)
There is heavy emphasis on counting, since it is used also in finding probabilities.