So in this case, n factorial was 7.

So 7 minus 3 is 4, so 4 factorial.

I think this will help.

If this was just 3 was so 7 times 6 times 5 times 4 times 3 times 2 times 1.

So it was 7 times 6 times 5 times 4 times 3 times 2 times 1.

I hung my hat on the peg.

25 plus let's see.

Let me see.

like this equation that we have up here.

Hang your hat on the hook.

It's actually a square, because it has the same length and the same width.

So we have a 1.

So the second term here, what am I going to do?

Multiplication

Multiplication

That seems to play out in all the research.

Sit down.

And finally for the last one I have two one times one five, so two by one, plus one by 5, which is equal to a 7, and so I get a 7 over there.

Then another 1 2.

So, you know, one times three is three.

Zero times 3 is zero.

So just to get the math here, so to get to this number right here, we just multiplied.

A cube with side x has length x, width x, and height x.

There's going to be equal to, right, 1, 3 times 0, 3 so that gives 1 times 0, plus 3 times 3.

This becomes 1 times 2 times 3 times up to N. And so using this product notation, this product from j equals 1 through n of this expression.

And now we just multiply both binomials, x times 2x, 2x squared.

Multiplication Puzzle

Right?

So we can factor this out to be 3 let me switch colors.

Times 2.

And actually, I'm going to be careful.

Sit down.

Be seated, Mikhail Ivanovich.

Sit down.

So this'll be a because I just distribute the a, right? a times a to the zero, plus a times a to the 1, plus a times a squared, plus all the way a times a to the n minus one, plus a times a to the n.

So it's just x times 2x, which is equal to 2x squared.

To get that x plus one out of the denominator, we multiply both sides of this equation times x plus one over one times this side.

And by the way, you can write multiplication, you know, either way.

Hang your coat on the hook.

Well this is a four by two matrix.

And what I mean by that is if you take three times five, that is equal to five times three and the ordering of this multiplication doesn't matter.

I don't. No one wears eyeglasses to bed.

Well, the chance of missing two shots in a row it's 4 .

And let me show that to you.

That's 32 times 42.