Using binomial theorem, evaluate: (99)^5 - Vidyadip

Question

Using binomial theorem, evaluate: $(99)^5$

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Answer

$(99)^5=(100-1)^5$
Using binomial theorem, we have
$(100-1)={ }^5 C_0(100)^5+{ }^5 C_1(100)^4(-1)+{ }^5 C_2(100)^3(-1)^2+{ }^5 C_3(100)^2(-1)^3$
$+{ }^5 C_4(100)(1)^4+{ }^5 C_5(-1)^5$
$=(100)^5+5(100)^4(-1)+10(100)^3+10(100)^2(-1)^3+5(100)+(-1)^5$
$=10000000000-500000000+10000000-100000+500-1$
$=10010000500-500100001$
$=9509900499$

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