( 5 0 )((1)/(4))^5+( 5 1 )((1)/(4))^4((3)/(4))+( 5 2 )((1)/(4

( 5 0 )((1)/(4))^5+( 5 ...

$\begin{aligned} & \left(\begin{array}{l}5 \\ 0\end{array}\right)\left(\frac{1}{4}\right)^5+\left(\begin{array}{l}5 \\ 1\end{array}\right)\left(\frac{1}{4}\right)^4\left(\frac{3}{4}\right)+\left(\begin{array}{l}5 \\ 2\end{array}\right)\left(\frac{1}{4}\right)^3\left(\frac{3}{4}\right)^2 \\ & +\left(\begin{array}{l}5 \\ 3\end{array}\right)\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^3+\left(\begin{array}{l}5 \\ 4\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^4+\left(\begin{array}{l}5 \\ 5\end{array}\right)\left(\frac{3}{4}\right)^5=?\end{aligned}$

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions raised to a power. It states that (x + y)^n can be represented as the sum of terms involving binomial coefficients and the powers of x and y. This theorem underpins the structure of the given expression by showing that if you add the terms together, you are computing (1/4 + 3/4)^5.

Binomial Coefficients

Binomial coefficients, denoted as 'n choose k', represent the number of ways to choose k elements from a set of n elements without regard to order. They are a central part of the Binomial Theorem, providing the multiplicative factors for each term in the expansion. In the provided expression, these coefficients determine the weights for each term in the sum.

Transcript

00:01 All right, so for the following question, we have a very long equation that we need to find the numerical value of.

00:08 So i'm going to write it out now.

00:10 This may take a little bit, since it is very long.

00:22 So obviously, as we can see, this is already a binomial expansion.

00:28 And hopefully, we all know the formula for that.

00:33 But if not, luckily, this question is so long that, you can probably find it by time i'm done writing this whole thing out.

00:44 You know, we're going to persevere.

00:47 I'm going to write it all out.

00:49 It's just easier when you write it out.

00:51 It's a better way of learning.

00:52 You should always write your questions out whenever doing it.

01:07 This one is a doozy.

01:11 Still going.

01:14 We'll be a couple more lines and we're going to be good.

01:16 Some of you can probably already kind of guess the rest of them, or not guess, sorry, know the rest of them.

01:22 Is a pattern or a formula or however you like to call it.

01:29 It should be a four.

01:31 And this is the last line.

01:36 So all this, we need to figure out what it equals.

01:43 So we know that from the binomial theorem, whenever we are given an x plus a y to the x to the y, n it can be equal to the sun of n j is equal it starts at zero of n over j and then we have x to the n minus j and then y to the j so when we actually expand that out in just purely mathematical form, it looks like this.

02:34 So n over 0, x to the n plus n over 1, x to the n minus 1, y plus n over 2, x to the n minus 1, y, plus n over 2, x to the 1, n minus 2, y to the power of 2.

02:59 And then plus and it goes on essentially until you reach n over n...

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