Algebra II
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California Standards Test: Algebra II
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California Standards Test: Algebra II (Graphing Inequalities
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CA Standards: Algebra II (Algebraic Division/Multiplication)
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CA Standards: Algebra II
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Algebra II: Simplifying Polynomials
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Algebra II: Imaginary and Complex Numbers
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Algebra II: Complex numbers and conjugates
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Algebra II: Quadratics and Shifts
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Examples: Graphing and interpreting quadratics
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Algebra ||: Conic Sections
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Algebra II: Circles and Logarithms
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Algebra II: Logarithms Exponential Growth
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Algebra II: Logarithms and more
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Algebra II: Functions, Combinatorics
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Algebra II: binomial Expansion and Combinatorics
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Algebra II: Binomial Expansions, Geometric Series Sum
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Algebra II: Functions and Probability
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Algebra II: Probability and Statistics
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Algebra II: Mean and Standard Deviation
Algebra II: binomial Expansion and Combinatorics 65 (done another way) - 66, combinatorics and binomial expansions
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- In the last video, we did problem 65,
- and we got the answer.
- And we did it in kind of an intuitive way, but if it was a
- slightly harder problem, it would've gotten a little more
- complicated, so I really wanted to show you how you can
- do it with the binomial theorem as well, or at least
- with binomial coefficients, and then we can try to see how
- they relate to each other, that they're
- actually the same thing.
- I realized that I should do this, because I'm looking at
- the next couple of questions, and they all deal with the
- binomial coefficient.
- So I figured I might as well do it the binomial way because
- that's probably what the California
- Standards want you to do.
- And it's a good way.
- It's good to be able to do it intuitively and with the
- binomial coefficients and understand how they're the
- same thing.
- So reading the same problem, Teresa and Julia are among 10
- students who have applied for a trip to D.C.
- Two students from the group will be selected at
- random for the trip.
- What is probability that Teresa and Julia will be the
- two students selected?
- So the way we did it last time, we said, OK, we could
- pick Julia first, and there's a 1 in 10 chance, and then we
- would pick Teresa, and that's a 1 in 9 chance.
- Or we could go the other way around, so it would be 2 times
- 1/10 times 1/9.
- 2, because there's two different ways to pick the
- people, and then for each of those ways, there's a 1 in 10
- chance of picking the first person and then a 1 in 9 for
- the second.
- That was the intuitive way we did that in the last video.
- The way you would do it with the binomial coefficient is
- you'd say, OK, I have 10 items, 10 students in this
- case, and I'm going to choose two of them.
- And people will often read this as 10 choose 2.
- And it's useful to memorize to some degree what this means.
- This means 10 factorial over 2 factorial
- times 10 minus 2 factorial.
- And so that is equal to-- and I'll give you in a second what
- the intuition is.
- That's equal to 10 factorial over 2
- factorial times 8 factorial.
- And at first, you might say, oh, my God, 10 factorial is a
- huge number.
- It's going to take me forever to multiply that out, until
- you realize that you're dividing it by 8 factorial.
- So 10 factorial, that's the same thing as 10 times 9 times
- 8 times 7, and you keep going.
- But 8 factorial is 8 times 7 times 6 and so
- forth and so on.
- So the 8 factorial, if you divide into 10 factorial,
- that's going to cancel out with that.
- And you're just left with 10 times 9.
- So if you have to say, how many different ways can you
- pick 2 people out of 10?
- You then get 10 times 9, which is 90, divided by 2 factorial,
- which is 2 times 1, so you say there's 45 different ways to
- pick 2 people.
- So the chances that I pick one of those ways, pick Julia and
- Teresa, is 1 out of 45.
- There are 45 combinations of picking 2 people out of 10.
- So the chances of picking any one of those
- combinations is 1 out of 45.
- So you get to the same answer.
- And just to get the intuition, this thing boils down
- to-- we just saw.
- It boils down to 10 times 9 over 2 factorial.
- And if you just want the intuition, there's 10 ways to
- pick the first person, there's 9 ways to pick the second
- person, and then there's 2 ways, or 2 factorial ways,
- that you could have picked one person and
- then another person.
- So it's the same intuition as this.
- But if you just wanted to kind of plug and chug and get the
- answer, you say, oh, 10 choose 2.
- That gives me the answer.
- I have 10 things.
- I need to choose 2.
- That gives me 45 different combinations and I want only
- one of them.
- So that's 1 out of 45.
- Next question.
- Let me see.
- Scroll up.
- All right.
- And when I saw this, I said, oh, they definitely want us to
- deal with binomial coefficients.
- And I'll show you two ways how to do this problem, too.
- So this one says 3y minus 1 to the fourth power.
- And you can multiply it out just using regular algebraic
- multiplication, polynomial multiplication.
- It will take you some time and you might
- make a careless mistake.
- I'm guessing they want us to use the binomial theorem.
- And the binomial expansion of this-- I'll use that
- combinatorics or that binomial coefficient notation.
- I think you'll get the pattern.
- And I've done several videos on this where I explain deeper
- why it works and all of that.
- But it equals 4 choose 0 times the first term to the fourth
- power, 3y to the fourth, times the second term to the 0th
- power, so you don't include it.
- That's times 1.
- Anything to the 0th power is 1.
- Fine.
- Plus 4 choose 1 times the first term, so that's 3y to
- the third power-- I'm just decrementing it on each term--
- Times the second term to the first power, so times negative
- 1 to the first power.
- And, in general, if you want to know what should this
- exponent be, it's going to be whatever is here.
- So this one is the same as that.
- So here we had a 0, so negative 1 to the 0, I didn't
- even take the trouble to write that down.
- Let me just keep going.
- I'll switch colors.
- So the next term, you could probably pause it and guess
- it, is going to be 4 choose 2.
- And I know you're wondering well, what is this 4 choose 1
- and 4 choose 2?
- Well, we saw it in the last problem, but I'll calculate
- them in a second.
- That's going to be 3y squared times negative 1 squared.
- And we have two more terms. Plus 4 choose 3
- times 3y to the 1.
- And just so you know, the exponents of these two should
- always add up to 4.
- So you have 4 and 0, you have 3 and 1, 2 and 2.
- So in this case, you have times negative 1 to the
- third-- and this 3 is that 3-- plus 4 choose 4 times negative
- 1 to the fourth power.
- So what do each of these binomial coefficients equal?
- 4 choose 0, that is equal to-- I'll do it in yellow.
- That's equal to 4 factorial over 0 factorial, which, you
- may or may not know, is defined to be 1, over 0
- factorial divided by 4 minus 0 factorial.
- And so that just turns into 1, right?
- 4 minus 0 is 4.
- So you get 4 factorial over factorial.
- So this is equal to 1.
- So let me write it out.
- So 3y the fourth.
- 3 to the fourth power is what?
- It's 81.
- So we get 81 y to the fourth.
- All right.
- Next term.
- Well, the first thing that you see is you have this negative
- 1 to the first power.
- This negative 1 is just going to keep flipping the sign.
- So negative 1 to the first power, that's just going to
- make this whole term negative and then we can
- ignore it from there.
- So we're going to get a negative there because of this
- negative 1 to the first power.
- And what's 4 choose 1?
- So that's 4 factorial over 1 factorial, which is also 1,
- divided by 4 minus 1 factorial.
- So that's 4 factorial over 3 factorial.
- So it's 4 times 3 times 2 times 1 divided by 3
- times 2 times 1.
- So that just equals 4.
- Try it out if you don't believe me.
- 4 times 3 times 2 times 1 divided by 3 times 2 times 1,
- you're just left with a 4.
- So you get 4.
- This coefficient becomes 4.
- But it's 4 times 3 to the third power.
- So maybe I'll just write it.
- The binomial coefficient is 4 times 27y to the third.
- And we already took care of the negative 1.
- Negative 1 to the first power is negative 1.
- And that's why I put the negative there.
- So the next term.
- We have a negative 1 to the second power, so it's going to
- be a positive.
- This is going to turn into a 1.
- That turns into a 1.
- So it's plus-- what's 4 choose 2?
- 4 choose 2 is-- I'm already running out of space.
- I'll write it up here.
- It's 4 factorial over 2 factorial
- times 4 minus 2 factorial.
- 4 minus 2 factorial, that's also 2 factorial.
- So it's equal to 4 times 3 times 2 times 1 divided by 2
- factorial is 2 and 2 factorial is 2.
- Let's say this 2 cancels out with that 2.
- This 2 cancels-- let's see, if you cancel out that 2, that
- becomes a 2.
- So you get 2 times 3 just becomes a 6.
- So that third term just becomes a 6.
- 4 choose 2 is 6.
- I know this is messy.
- So you have a plus 6 times 3y squared.
- So 9 times 9y squared.
- And we could go keep going.
- And actually, the binomial coefficient, the next one's
- going to be a 4, and it's negative 1 to the third, so
- it's going to be a minus 4 times 3y to the 1.
- I mean, we could keep going.
- Actually, let's just keep going.
- So the next term, 4 choose 3, I'll leave that for you to
- calculate it, but I already know that since we're kind of
- past the middle point, the binomial coefficient is going
- to be the same as 4 choose 1.
- So it's going to be a 4 and we have a minus 1 to the third,
- so this is minus 1.
- So it's minus 4 times 3y to the 1, so times 3y.
- And then we have the last term.
- 4 choose 4-- you could calculate it using this
- formula, but that turns out to be 1 as well-- plus 1 times
- negative 1 to the fourth, so that's just plus 1.
- And let's see what this is equal to.
- Let's see if we can simplify it.
- This is equal to 81y to the fourth minus-- let's see, 4
- times 20 is 80, 4 times 7 is 28, so it'd be
- 108, 108y to the third.
- 6 times 9 is 54, so plus 54y squared.
- Minus 12y plus 1.
- And if we look at the choices, only choice A
- is the right answer.
- And actually, we could have figured out it was choice A
- just by going one term deep, because that's where all of
- the choices- became different, so we could have
- saved a lot of work.
- Another option: Instead of evaluating each of the
- binomial coefficients, another option would have been to do a
- Pascal's triangle, where you say, OK, if I have a plus b to
- the first power, the coefficients are 1 and 1,
- right, 1a plus 1b.
- If I take it to the second power, a plus b squared, the
- coefficients become 1 plus 1 is 2.
- You bring down a 1 both places.
- And that makes sense because a plus b squared is 1a squared
- plus 2ab plus 1b squared.
- And then we could have kept going.
- This would have been to the third power.
- You had a 1, a 3, a 3 and a 1.
- And then the fourth power is what this problem was about.
- So you get a 1, you get a 4.
- 3 plus 3 is 6.
- 3 plus 1 is 4, and you get a 1.
- And so these would be the coefficients, and in this
- case, we were doing 3y minus 1 to the fourth power.
- If we had just drawn this out really fast, we would have
- said, well, this becomes 1 times 3y to the fourth plus 4
- times 3y to the third-- I just keep decrementing the 3y--
- times negative 1 to the 1 plus 6 times 3y squared--
- this 6 is this one.
- I'm getting a little misaligned-- 3y squared times
- negative 1 squared.
- So every time I increment the power on negative 1 and I
- decrement the power on 3y.
- And then plus 4.
- Now I'm at this one.
- Plus 4 times 3y negative 1 to the third.
- And then finally plus 1 times 3y to the 0 is just 1 times
- negative 1 to the the fourth power.
- And I would have evaluated it.
- And that might have been a slightly faster
- way of doing it.
- But I wanted to do this a couple of different ways
- because I think the binomial theorem and binomial expansion
- is something that people forget, and it's important to
- kind of have the neurons make a lot of cross-connections so
- that you don't forget it when you're 30 years old.
- Anyway, this was the least number of problems I've done
- in a video, but I think it was worth it.
- I'll see you in the next video.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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