B=a2_]-a2_^&c.,  =a2  taken  as  many  times  as  ^         ,  ^.  2

is  equal  to  the  No.  of  combinations  of  n  things>  ^==-^ — ^— •
taken  two  together,  which  is  (Art.  308),  5

C=a^-|-^^+»  ^c-j  ==^^  taken  as  many  ^
Umes  as  is  equal  to  the  No.  of  combinations  I  __w(?i — l){n — ^2)a*
of  the  things  taken  three  together,  which  is  |  1  •  2  •  3        '

(Art.  308),  J

&c.  =  &c.
K=aaa  ....  to  n  factors  =a".
Also,  {x-^a)(x-\-'b){x-{-c ) (^+Z)  becomes

{x-^d){x~\-aXx-\-d) {X'\-a)^{x-\-aY,

.-.    (.r+^)"=.rn+^...rn-l1-<^.^-l)..2.^n-2.^<n~l)(7l---2)^3         3

1-2  1-2-3

+ +«".
By  changing  a:  to  a  and  a  to  x,  we  have

(a+a?)"=a"+7ia»-ia:+^:^!Lrl)a'»-2a;2+
1-2  ^

^  n(?i~l)(7i— 2)  ,  ,  ,

1>2»3    ^     ^ +^"-


BINOMIAL  THEOREM.  267

Let  a=l,  then  since  every  power  of  1  is  1,

Cor.  1 . —  It  is  obvious  that  the  sum  of  the  exponents  of  a  and  x
in  each  term  =n.
Cor.  2. —  If  either  term  of  the  binomial  is  negative,  every  odd
power  of  that  term  will  be  negative  (Art.  193)  ;  therefore  the
signs  of  the  terms  in  which  the  odd  powers  are  found  will  be
negative.
^        ^  ^1-2  1.2-3         ^

Cor.  3 . —  The  general  term  of  the  series  is

^(^1)(^2) (n-^+2)^^_,^,^,

1.2.3        (r— 1)

For  the  1st  term  is a*»,

2nd    «      " nar-%

3rd    «      « 5^?Zi)a»-2a;2,

4th    "     «    nC7i~l)(7i~2)^„_3^3

1.2.3
&c.,  &c.
