Question

The coefficient of $x^{53}$ in the expansion of ${\sum }_{m=0}^{100}{{}^{100}C}_m{(x-3)}^{100-m}.2^m$ is

• A. ${{}^{100}C}_{47}$
• B. ${{}^{100}C}_{53}$
• C. $-{{}^{100}C}_{53}$
• D. $-{{}^{100}C}_{100}$

Answer 1

The correct option is C $-{{}^{100}C}_{53}$

The above is the general term of
$\left(\right(x-3)+2{)}^{100}$
$=(x-1{)}^{100}$
$=(1-x{)}^{100}$
Hence re-writing the general term we get
$T_{r+1}=(-1{)}^r{}^{100}{C}_r{x}^r$
Now the term consisting of $x^{53}$ is
$T_{54}=(-1{)}^{53}{}^{100}{C}_{53}$
$=-{}^{100}{C}_{53}$