Lesson 11 of 11 · 15 min

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Must-know facts

16 facts

1Multiplication principle: m ways then n ways gives m × n ways; separate cases add.
2Fill the most restricted place first, such as the units digit of an even number or a leading digit that cannot be 0.
3n! = 1 × 2 × … × n, with 0! = 1 and n! = n × (n − 1)!.
4nPr = n!/(n − r)!: ordered arrangements of r out of n different objects, no repetition.
5With repetition allowed, r places filled from n objects give nʳ arrangements.
6n objects with p₁, p₂, … alike: n!/(p₁! p₂! …) arrangements.
7nCr = n!/(r!(n − r)!): unordered selections of r out of n.
8nPr = nCr × r!: every selection can be ordered in r! ways.
9nCr = 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

Using nCr for a captain and vice-captain.

The two posts are different, so order matters: 12P2 = 132, not 12C2 = 66.

Using nPr for a team of 4 from 7.

A team has no order, so 7C4 = 35; 7P4 = 840 counts running orders.

Counting 3-digit numbers from 0 to 4 as 5 × 4 × 3 = 60.

Numbers cannot start with 0; subtract 4P2 = 12 to get 48.

Forgetting the arrangements inside a glued block.

RELAY with E and A together is 4! × 2! = 48, not 4! = 24.

Placing separated objects between the others only, not at the ends.

4 girls in a row leave 5 gaps including both ends, so the boys fill 5P3 places.

Dividing only by the count of one repeated letter.

BALLOON repeats L and O, so divide by 2! × 2!: 1260.

Adding in 'and then' problems or multiplying in 'or' problems.

Successive jobs multiply; separate cases add.

Writing 3! + 4! = 7!.

3! + 4! = 30, while 7! = 5040; factorials do not add this way.

Counting 'at least 2 girls' as 4C2 × 8C3.

That counts many teams more than once. Add the exact cases or subtract the unwanted ones: 186.

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.