Question

3. (2x 3)6

Answer 1

The given expression ( 2x−3 ) 6 .

The binomial expansion is given by

( 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 the expression of ( 2x−3 ) 6 and ( a+b ) n values of a=2x , b=−3 and n=6 .

Formula for Combination is given by,

C n r = n! r!( n−r )! ,0≤r≤n

According to the question

C 6 0 = 6! 0!( 6−0 )! = 6×5×4×3×2×1 ( 6×5×4×3×2×1 ) =1

Similarly all the values expanded in the same manner.

Substitute the values of a, b and n in equation (1), to expand the expression.

( 2x−3 ) 6 = C 6 0 ( 2x ) 6 ( 3 ) 0 − C 6 1 ( 2x ) 5 ( 3 )+ C 6 2 ( 2x ) 4 ( 3 ) 2 − C 6 3 ( 2x ) 3 ( 3 ) 3 + C 6 4 ( 2x ) 2 ( 3 ) 4 − C 6 5 ( 2x ) ( 3 ) 5 + C 6 6 ( 3 ) 6 =64 x 6 −6( 32 x 5 )( 3 )+15( 16 x 4 )( 9 )−20( 8 x 3 )( 27 )+15( 4 x 2 )( 81 ) −6( 2x )( 243 )+729 =64 x 6 −576 x 5 +2160 x 4 −4320 x 3 +4860 x 3 −2916x+729

Thus the expansion of the ( 2x−3 ) 6 is 64 x 6 −576 x 5 +2160 x 4 −4320 x 3 +4860 x 3 −2916x+729 .