Question

How do you expand the binomial $ {(x + 4)^5} $ using the binomial theorem?

Answer 1

Hint: The binomial expansion or the binomial theorem describes the algebraic expansion of the powers of the binomial (binomial is the pair of two terms). Use formula $ {(a + b)^n} = {}^n{C_a}{a^n} + {}^n{C_1}{a^{n - 1}}{b^1} + ..... $ for binomial expansion. Where, $ {}^n{C_a} $ represents the total number of possible ways and use of the laws of powers and exponent accordingly.

Complete step by step solution:
By using the formula of the binomial expansion –
 $ {(a + b)^n} = {}^n{C_a}{a^n} + {}^n{C_1}{a^{n - 1}}{b^1} + ..... $
Here, $ a = x,b = 4,n = 5 $
And using $ ^nc{}_r = \dfrac{{n!}}{{r!(n - r)!}} $
 $ 5{C_0} = 1,\,5{C_1} = 5,\;5{C_2} = 10,\;5{C_3} = 10,\;5{C_4} = 5,\;5{C_1} = 1 $
Now, take given binomial expansion and apply the above formula in it –
 $ {(x + 4)^5} = {x^5} + 20{x^4} + 160{x^3} + 640{x^2} + 1280x + 1024 $
So, the correct answer is “ $ {(x + 4)^5} = {x^5} + 20{x^4} + 160{x^3} + 640{x^2} + 1280x + 1024 $”.

Note: Know the difference between the permutations and combinations and apply its formula accordingly. In permutations, specific order and arrangement is the most important whereas a combination is used if the certain objects are to be arranged in such a way that the order of objects is not important.
Formula for combinations - $ ^nc{}_r = \dfrac{{n!}}{{r!(n - r)!}} $
Formula for the permutations - $ {}^np{}_r = \dfrac{{n!}}{{(n - r)!}} $