What is the fifth term in the binomial expansion of (x - 3y)^8?

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To determine the fifth term in the binomial expansion of ((x - 3y)^8), we can use the Binomial Theorem, which states that the expansion of ((a + b)^n) can be expressed as:

[

\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

]

In this case, (a = x), (b = -3y), and (n = 8).

The general term in the expansion, often denoted as (T_{k+1}), is given by:

[

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

]

For the fifth term, we set (k = 4) (since we start counting from (k = 0)). Thus, we have:

[

T_{5} = \binom{8}{4} (x)^{8-4} (-3y)^4

]

Now, calculating each part:

  1. Calculate the binomial coefficient:

(\binom{8}{4} =

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