The formal expression of the Binomial Theorem is as follows:
#(a+b)^n=sum_"k=0"^n((n!)/(k!(n-k)!))a^"n-k"b^k#
Given : #a = s, b = 2x# & #n = 5&
#(s+2x)^n=sum_"k=0"^5((5!)/(k!(5-k)!))s^"5-k"(2x)^k#
When k = 0, R H S becomes
#((5!)/(0! * 5!))s^5 * (2x)^0 = s^5#
When k = 1, R H S becomes
#((5!)/(1! * (5-1)!))s^4 * (2x)^1 = 10 s^4 x#
When k = 2, R H S becomes
#((5!)/(2! * (5-2)!))s^3 * (2x)^2 = 40 s^3 x^2#
When k = 3, R H S becomes
#((5!)/(3! * (5-3)!))s^2 * (2x)^3 = 80 s^2 x^3#
When k = 4, R H S becomes
#((5!)/(4! * (5-4)!))s^1 * (2x)^4 = 80 s x^4#
When k = 5, R H S becomes
#((5!)/(5! * (5-5)!))s^0 * (2x)^5 = 32 x^5#
Expansion of #(s + 2x)^5 = s^5 + 10 s^4 x + 40 s^3 x^2 + 80 s^2 x^3 + 80 s x^4 + 32 x^5#
