Iwillsueyou69420

is there a way to do it without arcos?

This means that these do not happen with 10 flips. 

Firstly, we calculate the probability that there are 3 heads in a row

Ways to have 3 heads in a row occur 8 times 1

7 times: 

6 times:

5 times:

4 times:

3 times:

2 times:

1 time:

find the amount of ways then multiply by 1/8

Then do the same for 4 tails in a row. Then substract it all from 1

First, we list the numbers each number can be next to

Note no number can be next to itself, and every number other than the ones on the side are next to two values

1: 5,6

2: 3,4,5,6

3: 2,4,5,6

4: 2,3,5,6

5: 1,2,3,4,6

6: 1,2,3,4,5

We realize that every number can be next to any value except 1, and only 1 cannot be next to most numbers. We have two cases

1. 1 is in the middle (not on the edge)

1 must be next to 5 and 6, so we make a group of 1,5,6, where 1 must be in the middle, which has 2 arrangements within. Outside there are 4! = 24 arrangements, making 24 x 2 = 48

2. 1 is on the edge

1 must be next to 5 or 6, so there are 2 possibilities for that. The 1 can also be on either edge, making another 2 ways. The other 4 have 4! = 24 ways, therefore there are 2 x 2 x 24 = 96

Therefore, the total ways is 48 + 96 = ... 

I'm going to let you do the addition yourself

Just kidding it is 144

firstly, you realize that \(a_n = 1+1/n(n+1)\), which is \((n^2+n+1)/n^2+n\) No idea what to do next, just a idea!

assuming you want a circle whose area is pi square units, the area of the hexagon is 6x area of a side length 1 equalateral triangle =ex 3sqrt3/2, so the ratio  circle area/hexagon area is 2pi/3sqrt3. 

Something that might help: 

Cauchy-Schwarz inequality states that

(∑aibi)2≤(∑ai2)(∑bi2)open paren sum of a sub i b sub i close paren squared is less than or equal to open paren sum of a sub i squared close paren open paren sum of b sub i squared close paren

(𝑎𝑖𝑏𝑖)2≤(𝑎2𝑖)(𝑏2𝑖)

.

a square foot is 0.3048 x 0.3048 = 0.09290304 square meters. The wall is 1440 square meters. A gallon covers 0.09290304 x 150 = 13.935456 square meters. So just divide 1440 by 13.935456, which is between 103 and 104(around 103 1/3), so the answer is 104

Or 133 and 79 are relatively prime, so 2