To expand any binomial of this kind, first write a^5, the first term.
The coefficient of this term is 1 and it is the first term. The
coefficient of the second term (which has a^4 and (2b)^1 in it) is the
exponent of a in the first term times the existing coefficient (1),
divided by the number of the term. So, the coefficient of the second
term is 5*1/1 = 5.
Okay, so now we have:
a^5 + 5*a^4*(2b)^1
To get the coefficient of the third term, apply the same rule. Take
the exponent of a in the second term, multiply by the coeffcient of
the second term and divide by the number of the term. So, 4*5/2 = 10.
Now we have:
a^5 + 5*a^4*(2b)^1 + 10*a^3*(2b)^2
Next, 3*10/3 = 10:
a^5 + 5*a^4*(2b)^1 + 10*a^3*(2b)^2 + 10*a^2*(2b)^3
Next, 2*10/4 = 5, and so on.