Question

(1-2r)5

Answer 1

The given expression is ( 1−2x ) 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 the expression of ( 1−2x ) 5 and ( a+b ) n values of a=1 , b=−2x and n=5 .

Formula for Combination is given by,

C n r = n! r!( n−r )! ,0≤r≤n

According to the question

C 5 0 = 5! 0!( 5−0 )! = 5×4×3×2×1 ( 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

( 1−2x ) 5 = C 5 0 ( 1 ) 5 − C 5 1 ( 1 ) 4 ( 2x )+ C 5 2 ( 1 ) 3 ( 2x ) 2 − C 5 3 ( 1 ) 2 ( 2x ) 4 − C 5 5 ( 2x ) 5 =1−5( 2x )+10( 4 x 2 )−10( 8 x 3 )+5( 16 x 4 )−( 32 x 5 ) =1−10x+40 x 2 −80 x 3 +80 x 4 −32 x 5

Thus the expansion of the ( 1−2x ) 5 is 1−10x+40 x 2 −80 x 3 +80 x 4 −32 x 5 .