1 1 counting techniques

PART 1.1: COUNTING TECHNIQUES X.1

FIGURATE NUMBERS: 2D Cases

n(n + 1) . 2 n 2 = Tn + Tn−1

Triangular Number

Tn =

Square Number

Tn = n 2 n(3n − 1) Tn = 2 2n(2n − 1) Tn = 2 2 5n − 3n Tn = 2 2 Tn = 3n − 2n

Pentagonal Number Hexagonal Number Heptagonal number Octagonal number Nonagonal number Decagonal Number Centered triangular number Centered square number Centered pentagonal number

Centered hexagonal number

n(7 n − 5) 2 2 Tn = 4n − 3n

Tn =

3n 2 + 3n + 2 2 2 2 Tn = n + (n − 1)

Tn =

5(n − 1) + 5(n − 1) + 2 Tn = 2 3 3 Tn = n − (n − 1) 2

Tn = 3n(n − 1) + 1

Centered heptagonal number Centered octagonal number

Centered nonagonal number Centered decagonal number

7n 2 − 7n + 2 Tn = 2 2 Tn = (2n − 1)

Tn = 4n 2 − 4n + 1 (3n − 2)(3n − 1) Tn = 2 2 Tn = 5(n − n ) + 1 3D Cases

Tetrahedral Number Centered cube number Octahedral number Dodecagonal Number

n + 3n 2 + 2n 6 3 3 Tn = n + (n + 1) Tn =

3

(

)

n 2n 2 + 1 Tn = 3 2 Tn = n + 4(n 2 − n )

Square Pyramid Number Pentagonal pyramidal number Hexagonal pyramidal number Heptagonal pyramidal number

X.2

2n 3 + 3n 2 + n 6 2 n (n + 1) Tn = 2 n(n + 1)(4n − 1) Tn = 6 n(n + 1)(5n − 2 ) Tn = 6 Tn =

PERMUTATIONS & COMBINATIONS:

Permutations:

= n!

Permutations (with repeats):

=

Ordered Combinations: Unordered Combinations:

n! (groupA)!×(groupB )!×... n! = (n − p )! n n! =   =  p  p!(n − p )!

Ordered Repeated Combinations: = n p Unordered Repeated Combinations: = Grouping:

X.3

( p + n − 1)! p!×(n − 1)!

 n  n − n1  n − n1 − n2  n! ... =  =   n3 n1!n2 !n3!...nr !  n1  n2  

MISCELLANEOUS:

Total Number of Rectangles and Squares from an a x b rectangle: ∑ = Ta × Tb

= TL −1 c(c + 1) Max number of pizza pieces: = +1 2 c(c + 3) Max pieces of a crescent: = +1 2 c 3 + 5c Max pieces of cheese: = +1 6 l (3l + 1) Cards in a card house: = 2 n− d Different arrangement of dominos: = 2 × n! Number of Interpreters:

b  a − MOD   a 1 a Unit Fractions: = + b b    b INT   + 1 b INT   + 1 a a   Angle between two hands of a clock: θ = 5.5m − 30h

Winning Lines in Noughts and Crosses: = 2(a + 1) P Bad Restaurant Spread: = 1− s n n 1   1 + 5   1 − 5     −  Fibonacci Sequence: = 5   2   2     ABBREVIATIONS (5.1, 5.2, 5.3, 5.4, 5.5) a=side ‘a’ b=side ‘b’ c=cuts d=double dominos h=hours L=Languages l=layers m=minutes n= nth term n=n number P=Premium/Starting Quantity p=number you pick r=number of roles/turns s=spread factor T=Term θ=the angle

X.4

THE DAY OF THE WEEK:

This only works after 1753

  31m   y   y   y   = MOD7 d + y +  + − +    12   4  100   400    d=day m=month y=year