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Question: How do you expand the binomial \( {(x + 4)^5} \) using the binomial theorem?...

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

Explanation

Solution

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=nCaan+nC1an−1b1+.....{(a + b)^n} = {}^n{C_a}{a^n} + {}^n{C_1}{a^{n - 1}}{b^1} + ..... for binomial expansion. Where, nCa{}^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=nCaan+nC1an−1b1+.....{(a + b)^n} = {}^n{C_a}{a^n} + {}^n{C_1}{a^{n - 1}}{b^1} + .....
Here, a=x,b=4,n=5a = x,b = 4,n = 5
And using ncr=n!r!(n−r)!^nc{}_r = \dfrac{{n!}}{{r!(n - r)!}}
5C0=1, 5C1=5,  5C2=10,  5C3=10,  5C4=5,  5C1=15{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=x5+20x4+160x3+640x2+1280x+1024{(x + 4)^5} = {x^5} + 20{x^4} + 160{x^3} + 640{x^2} + 1280x + 1024
So, the correct answer is “ (x+4)5=x5+20x4+160x3+640x2+1280x+1024{(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 - ncr=n!r!(n−r)!^nc{}_r = \dfrac{{n!}}{{r!(n - r)!}}
Formula for the permutations - npr=n!(n−r)!{}^np{}_r = \dfrac{{n!}}{{(n - r)!}}