Using binomial theorem, evaluate: (102)^5 - Vidyadip

Question

Using binomial theorem, evaluate: $(102)^5$

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Answer

$(102)^5 = (100 + 2)^5$
Using binomial theorem, we have
${(100 + 2)^5}{ = ^5}{C_0}{(100)^5}{ + ^5}{C_1}{(100)^4}(2)$${ + ^5}{C_2}{(100)^3}{(2)^2}$
${ + ^5}{C_3}{(100)^2}{(2)^3}{ + ^5}{C_4}(100){(2)^4}$${ + ^5}{C_5}{(2)^5}$
$= (100)^5 + 5(100)^4(2) + 10(100)^3(2)^2 + 10(100)^2(2)^3 + 5(100)(2)^4 + (2)^5$
$= 10000000000 + 1000000000+ 40000000 + 800000 + 8000 + 32$
$= 11040808032$

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