Find the sixth term of the expansion of $$ (x+6)^{7} $$

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To find the sixth term of the expansion of $$ (x+6)^{7} $$, we can use the binomial theorem. The general formula for the $$ k $$th term of the expansion of $$ (a+b)^{n} $$ is given by:

$$ T_{k+1} = \binom{n}{k} \cdot a^{n-k} \cdot b^{k} $$

In this case, $$ a = x $$, $$ b = 6 $$, $$ n = 7 $$, and we want to find the sixth term, which corresponds to $$ k = 5 $$.

Substitute the values into the formula:

$$ T_{6} = \binom{7}{5} \cdot x^{7-5} \cdot 6^{5} $$

Now, we can calculate the value of the sixth term.
Simplify the expression by following steps:
- step0: Solution:
  $$ { }_{7}C_{5}x^{7-5}\times 6^{5}$$
- step1: Subtract the numbers:
  $$ { }_{7}C_{5}x^{2}\times 6^{5}$$
- step2: Expand the expression:
  $$\frac{7!}{5!\times \left(7-5\right)!}\times x^{2}\times 6^{5}$$
- step3: Subtract the numbers:
  $$\frac{7!}{5!\times 2!}\times x^{2}\times 6^{5}$$
- step4: Reduce the fraction:
  $$\frac{7\times 6}{2!}\times x^{2}\times 6^{5}$$
- step5: Multiply the numbers:
  $$\frac{42}{2!}\times x^{2}\times 6^{5}$$
- step6: Divide the terms:
  $$21x^{2}\times 6^{5}$$
- step7: Multiply the numbers:
  $$163296x^{2}$$
The sixth term of the expansion of $$ (x+6)^{7} $$ is $$ 163296x^{2} $$.
The sixth term is $$ 163296x^{2} $$.

Find the sixth term of the expansion of \( (x+6)^{7} \)

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