Question

Using binomial theorem, evaluate the following:
${\left(102\right)}^5$

Answer 1

${\left(102\right)}^5=(100+2{)}^5$

Using binomial theorem, we have

$(100+2{)}^5=5_{C_0}(100{)}^5+5_{C_1}\left(100{)}^4\right(2)+5_{C_2}(100{)}^3(2{)}^2+5_{C_3}(100{)}^2(2{)}^3+5_{C_4}(100\left)\right(2{)}^4+5_{C_5}(2{)}^5$

=$(100{)}^5+5(100{)}^4\left(2\right)+10\left(100{)}^3\right(2{)}^2+10\left(100{)}^2\right(2{)}^3+5\left(100\right)(2{)}^4+(2{)}^5$

$=10000000000+1000000000+40000000+800000+8000+32$

$=11040808032$