APPENDIX 2

Trian gul ar Numbers and
Combinations

The identity *C, is used to express the number of different
combinations which may be made of » things taken x at a
time, and is calculated according to the following expansion:
ety wrn) )
(x—1) (x—2) (*—3)
The number of factors in both denominator and numerator
is equivalent to x. Thus, the number of combinations of g
things taken g at a time (that is, n=9; ¥=3)

__g><8><7_8
A aR

The expansion "C, displays a relationship between itself
and the triangular numbers referred to in Chapter 2. This
relationship arises out of the fact that the number of com-
binations of any number of different things taken #wo at a
time will always be a triangular number. The expansions
for different values of n are as follows:

n= 2 1l R
Expansiony Mo 800 My V8L i Gy
Combinations: 1 B R e

We have previously noted that triangular numbers are
equal to the sums of series of consecutive integers from 1 up-
wards. Thus:

142 = 3

i+24+3 =06

1+2+3+4=1I0
165