Lesson 11 of 11 · 15 min
Chapter review
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Must-know facts
16 facts
- 01Multiplication principle: m ways then n ways gives m × n ways; separate cases add.
- 02Fill the most restricted place first, such as the units digit of an even number or a leading digit that cannot be 0.
- 03n! = 1 × 2 × … × n, with 0! = 1 and n! = n × (n − 1)!.
- 04nPr = n!/(n − r)!: ordered arrangements of r out of n different objects, no repetition.
- 05With repetition allowed, r places filled from n objects give nʳ arrangements.
- 06n objects with p₁, p₂, … alike: n!/(p₁! p₂! …) arrangements.
- 07nCr = n!/(r!(n − r)!): unordered selections of r out of n.
- 08nPr = nCr × r!: every selection can be ordered in r! ways.
- 09nCr = nC(n − r); if nCa = nCb then a = b or a + b = n.
- 10Pascal's rule: nCr + nC(r − 1) = (n + 1)Cr.
- 11Together: glue into a block and multiply by the arrangements inside it.
- 12Apart: arrange the others first, then place the separated objects in the gaps.
- 13Not together = total − together.
- 14'At least' selections: add the cases, or subtract the unwanted cases from the total.
- 15Select first, then arrange: choose with nCr, then multiply by r!.
- 16Sports-day numbers: 12 kits, 720 lock codes without repeats, 132 captain pairs, 35 relay teams and 840 line-ups.
Common traps
Where marks are lost
Formulas
12 to know
Multiplication principle
ways = m × n × p × …
For jobs done one after another.
Factorial
n! = 1 × 2 × 3 × … × n; 0! = 1
n! = n × (n − 1)!
Permutations, no repetition
nPr = n!/(n − r)! = n(n − 1)…(n − r + 1)
0 ≤ r ≤ n; nPn = n!, nP0 = 1.
Permutations with repetition
n × n × … × n (r times) = nʳ
Each place has all n choices.
Repeated objects
n!/(p₁! p₂! … pₖ!)
p₁, p₂, … objects alike of each kind.
Combinations
nCr = n!/(r!(n − r)!)
0 ≤ r ≤ n; nC0 = nCn = 1.
Link
nPr = nCr × r!
Choose, then order.
Symmetry
nCr = nC(n − r)
Choosing r to take = choosing n − r to leave.
Equal combinations
nCa = nCb ⇒ a = b or a + b = n
Used to find n.
Pascal's rule
nCr + nC(r − 1) = (n + 1)Cr
1 ≤ r ≤ n.
Together (block)
(n − k + 1)! × k!
k different objects kept together among n different objects.
Apart (gaps)
m! × (m + 1)Pk
k different objects, no two together, placed among m others.
Key terms
12 terms
- Multiplication principle
- If one job can be done in m ways and then another in n ways, both can be done in m × n ways.
- Factorial
- n! is the product of the first n natural numbers, with 0! = 1.
- Permutation
- An arrangement of objects in a definite order.
- nPr
- How many ordered arrangements of r things can be made from n different things.
- Combination
- A selection of objects in which the order does not matter.
- nCr
- How many unordered selections of r things can be made from n different things.
- Repetition allowed
- An object may be used again in later places.
- Alike objects
- Objects that cannot be told apart, so swapping them makes no new arrangement.
- Block method
- Treating objects that must stay together as one unit.
- Gap method
- Placing objects that must be apart in the gaps between the others.
- Pascal's rule
- nCr + nC(r − 1) = (n + 1)Cr.
- Dictionary order
- Listing words by first letter, then second letter, and so on, in alphabetical order.