Here we can find the coefficient of x53 in the given expression We know that Binomial theorem for positive integral index If a , b are any two real numbers and n any natural number then (a+b)n =
n
‌
C0anb° +
n
‌
C1an−1b +
n
‌
C2an−2b2 + ... +
n
‌
Cran−rbr + ... +
n
‌
Cnx0bn , where
n
‌
Cr =
n!
(n−r)!r!
Here
n
‌
C0,
n
‌
C1,
n
‌
C2 ...
n
‌
Cn are called binomial coefficients i.e., (a+b)n =
n
Σ
r=0
n
‌
Cran−rbr Given
100
Σ
m=0
100Cm(x−3)100−m.2m On expanding the summation , we have , = 100C0(x−3)100+100C1(x−3)99.2 + 100C2(x−3)98.22 + ... + 100C100.2100 = [(x−3)+2]100 = (x−1)100 = (1−x)100 (Since, the power is even, by binomial expansion) Therefore, coefficient of x53 = (−1)53.
100
‌
C53 = −
100
‌
C53 Hence, The coefficient of x53 in the expression of
100
Σ
m=0
100Cm(x−3)100−m.2m = −
100
‌
C53 another method : we have
100
Σ
m=0
100Cm(x−3)100−m.2m = [(x−3)+2]100 = (1−x)100 Therefore, coefficient of x53 =