MHB George's question at Yahoo Answers regarding the binomial theorem

AI Thread Summary
The discussion addresses George's question about identifying the term containing x^36 in the expansion of (x^6 + 2)^{18}. Using the binomial theorem, the relevant term is derived from the expression, leading to the equation 6(18-k) = 36. Solving this gives k = 12, indicating that the term is 18 choose 12 multiplied by x^36 and 2^12. The final result for the term containing x^36 is 76038144x^36. This demonstrates the application of the binomial theorem in polynomial expansions.
MarkFL
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Here is the question:

Which term of the expansion of (x^6 + 2 )^{18} contains x^{36}?

Please explain
thank you

I have posted a link there to this thread to the OP can view my work.
 
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Hello George,

By the binomial theorem, we have:

$$\left(x^6+2 \right)^{18}=\sum_{k=0}^{18}\left[{18 \choose k}\left(x^6 \right)^{18-k}2^k \right]=\sum_{k=0}^{18}\left[{18 \choose k}x^{6(18-k)}2^k \right]$$

Hence, the term which contains $x^{36}$ will be the term for which:

$$6(18-k)=36$$

$$18-k=6$$

$$k=12$$

And so, this term is:

$${18 \choose 12}x^{36}2^{12}=18564\cdot x^{36}\cdot 4096=76038144x^{36}$$
 
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