1. xi(x 3)8
The given expression is ( x + 3 ) 8 we have to find the coefficient of the x 5 .
The formula for binomial expansion is ,
( a + b ) n = C n 0 a n + C n 1 a n − 1 b + C n 2 a n − 2 b 2 + .......... + C n n − 1 a . b n − 1 + C n n b n (1)
On comparing both the expression ( x + 3 ) 8 and ( a + b ) n ,value of a = x , b = 3 and n = 8 .
Let ( r + 1 ) t h terms contains coefficient of x 5 in expansion of ( x + 3 ) 8 .
General terms expressed as
T r + 1 = C n r a n − r ( b ) r T r+1 = C 8 r ( x ) 8 − r ( 3 ) r (2)
On comparing equation (2) power of x with the power of x 5
8 − r = 5 r = 3
Substitute the value of r = 3 in equation (2),
T 3 + 1 = C 8 3 ( x ) 8 − 3 ( 3 ) 3 = C 8 3 ( x ) 5 ( 3 ) 3
Coefficient of x 5 is equal to C 8 3 ( 3 ) 3 ,
On solving value of coefficient of x ,
C 8 3 ( 3 ) 3 = 8 ! 3 ! ( 8 − 3 ) ! ( 3 ) 3 = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 ( 3 × 2 × 1 ) ( 5 × 4 × 3 × 2 × 1 ) ( 3 ) 3 = 56 × 27 = 1512
Thus the coefficient of x 5 is 1512 .
The given expression is
The formula for binomial expansion is ,
On comparing both the expression
Let
General terms expressed as
On comparing equation (2) power of
Substitute the value of
Coefficient of
On solving value of coefficient of
Thus the coefficient of
