Using Binomial Theorem, evaluate $(101)^{4}$

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Solution

$(101)^{4} = (100 + 1)^{4}$
$=^{4} C_{0} (100)^{4} +^{4} C_{1} (100)^{3} (1) +^{4} C_{2} (100)^{2} (1)^{2} +^{4} C_{3} (100) (1)^{3} +^{4} C_{4} (1)^{4}$
$= (100)^{4} + 4 (100)^{3} + 6 (100)^{2} + 4 (100) + (1)^{4}$
$= 100000000 + 4000000 + 60000 + 400 + 1$
$= 104060401$

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