Question

Using Binomial Theorem, evaluate \((99)^5\).

Answer 1

\(99\) can be written as the sum or difference of two numbers whose powers are easier to calculate and then, Binomial Theorem can be applied.
It can be written that, \(99 = 100 -1\)
∴\((99)5 = (100-1)5\)
=\(^5C_0(100)^5 -^5C_1(100)^4(1)+ ^5C_2(100)^3(1)^2 -^5C_3(100)^2 (1)^3 +^5C_4 (100) (1)^4 - ^5C_5(1)^5\)
=\((100)^5 - 5(100)^4+10(100)^3 -10(100)^2+5(100)-1\)
=\(10000000000-500000000+10000000-100000+500-1\)
=\(10010000500-500100001\)
=\(9509900499\)