Free QCE Year 11 Specialist Mathematics Flashcard Review
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State the multiplication principle.
If one stage can be done in m ways and the next in n ways, together they can be done in m × n ways.
When do you use the addition principle?
When the alternatives are mutually exclusive (cannot both happen): add the counts.
Inclusion–exclusion for two sets?
|A ∪ B| = |A| + |B| − |A ∩ B|.
What is 0! and why?
0! = 1, so that formulas such as ⁿPₙ = n!/0! remain consistent.
Formula for ⁿPᵣ?
ⁿPᵣ = n!/(n − r)!, the number of ordered selections of r from n distinct objects.
Formula for ⁿCᵣ?
ⁿCᵣ = n!/(r!(n − r)!), the number of unordered selections.
Arrangements of n objects with p alike and q alike?
n!/(p! q!).
How do you count arrangements with certain objects together?
Treat them as one block, arrange the blocks, then multiply by the internal arrangements.
Why is ⁿCᵣ = ⁿCₙ₋ᵣ?
Choosing r to include is the same as choosing n − r to exclude.
How do you select from multiple groups?
Multiply the combinations for each group, e.g. ⁵C₂ × ⁷C₃.
Inclusion–exclusion for three sets?
|A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|.
Quick way to find P(at least one)?
1 − P(none).
When is the implication P ⇒ Q false?
Only when P is true and Q is false.
Converse of P ⇒ Q?
Q ⇒ P. It may or may not be true.
Contrapositive of P ⇒ Q?
¬Q ⇒ ¬P, which is logically equivalent to P ⇒ Q.
Negation of P ⇒ Q?
P and ¬Q.
What does P ⟺ Q mean and how is it proved?
P if and only if Q; prove both P ⇒ Q and Q ⇒ P.
Negation of ∀x, P(x)?
∃x such that ¬P(x).
Define ℚ and ℚ′.
ℚ: numbers a/b with a, b ∈ ℤ, b ≠ 0. ℚ′: real numbers that are not rational (irrationals).
Is 0 in ℤ⁺?
No. 0 ∈ ℤ, but 0 is in neither ℤ⁺ nor ℤ⁻.