How do you use the Binomial Theorem to expand #(x + y)^5#?

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Feb 10, 2016

Answer:

The final answer :
#(a+b)^5=a^5+5.a^4.b+10.a^3.b^2+10.a^2.b^3+5.a^1.b^4+b^5#

Explanation:

The binomial theorem tells us that if we have a binomial (a+b) raised
to the #n^(th)# power the result will be

#(a+b)^n=sum_(k=0)^nc_k^n *a^(n-k)*b^(n)#

where #" "c _k^n= (n!)/(k!(n-k)!)#

and is read "n CHOOSE k equals n factorial divided by k factorial (n-k) factorial".

So #(a+b)^5=a^5+5.a^4.b+10.a^3.b^2+10.a^2.b^3+5.a^1.b^4+b^5#

we notice that the powers of ' a ' keeps decreasing from 5 (which representes ' n ') until it reaches #a^("zero")# at the last term.

also we notice that the power of ' b ' keeps increasing from zero untill it reaches 5 at the last term.

Now the we have to determine the coefficient of each term through the...

#c_k^n= (n!)/(k!(n-k)!)#

first coefficient #c_0^5=(5!)/(0! .5!)=1#

second #c_1^5=(5!)/(1! .4!)=5#

#c_2^5=(5!)/(2! .3!)=10#

#c_3^5=(5!)/(3! .2!)=10#

#c_4^5=(5!)/(4! .1!)=5#

#c_5^5=(5!)/(5!.0!)=1#

but the calculation of combinations can be tedious..so fortunately
there is an awesome way to determine the binomial coefficients which is Pascal's triangle

enter image source here

it is easy to deduce this triangle :

hope that helps ! : )

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