All Answers 353121 Answers

there are 22, $1 bills since we get the equation x+2(x+11)+3(x-18)=100 where x=22 there so there are 22, $1 bills

It is generally much easier to compute the "combinations" first and then convert the combination into permutations. With a very short computer code, each of these combinations is converted to permutations and then summed up. I wrote such a short computer code just for the pupose of converting these 71 combinations into permutations and the result is: 2,111 permutations.

12233 , 12234 , 12235 , 12244 , 12245 , 12255 , 12333 , 12334 , 12335 , 12344 , 12345 , 12355 , 12444 , 12445 , 12455 , 12555 , 13334 , 13335 , 13344 , 13345 , 13355 , 13444 , 13445 , 13455 , 13555 , 14444 , 14445 , 14455 , 14555 , 15555 , 22333 , 22334 , 22335 , 22344 , 22345 , 22355 , 22444 , 22445 , 22455 , 22555 , 23334 , 23335 , 23344 , 23345 , 23355 , 23444 , 23445 , 23455 , 23555 , 24444 , 24445 , 24455 , 24555 , 25555 , 33344 , 33345 , 33355 , 33444 , 33445 , 33455 , 33555 , 34444 , 34445 , 34455 , 34555 , 35555 , 44445 , 44455 , 44555 , 45555 , 55555 , Total = 71 combinations.

We're going to turn each sentence into an equation.

a = number of $1, b= number of $2, c= number of $3

b=a+11

c=a-18

a+2a+3b=100

Substituting the first two equations into the third equation,

a+2a+22+3a-54=100

Simplifying, we get 6a=132, a=22

So there are 22 one-dollar bills.

s\o we have 65-60=5 and 60*10=600 then we do 600/5=120 which means 2 hours later at 5:10

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.

For example, 10^3 = 1,000; therefore, log10 1,000 = 3.

1/2 base * hieght is the formula we use to solve th area

my username on aops is amazingxin777 (same as this one)

Hint

we get 180-3angle A=AngleB

and 

AB=BC-2

Just continue on the sequence

\(1\text {st} \space \text {term}\)      \(2\text {nd} \space \text {term}\)       \(3\text{rd}\space\text{term}\)       \(4\text{th} \space \text{term}\)        \(5\text{th} \space \text{term}\)        \(6\text{th} \space\text{term}\)        \(7\text{th} \space \text {term}\)      \(8\text {th} \space \text{term}\)      \(9\text{th} \space \text{term}\)      

20 (even)      10 (even)          5 (odd)            16 (even)           8 (even)           4 (even)           2 (even)         1 (odd)           4 (even)

\(10\text{th} \space\text{term}\)

         2 

So the answer is \(\boxed {2}\)

hint: \(\text {distance} =\) \(\text {speed} \times \text {time}\)

Alright, let's write a couple of equations.

First, let x equal the speed at which she went to the other town (what we're trying to find) and let y equal the time it takes her.

Then, because distance=speed*time, 150=xy

Our second equation is going to be on her way back. 150=(x+15)(y+\(\frac{1}{2}\)). We put \(\frac{1}{2}\)hr instead of 30 minutes because our speeds are given in km/hrs. 

Now we expand: 150=xy+\(\frac{1}{2}\)x+15y+\(\frac{15}{2}\)

Substitute 150=xy (our first equation). 150=150+\(\frac{1}{2}\)x+15y+\(\frac{15}{2}\)

This simplifies to \(\frac{1}{2}\)x+15y+\(\frac{15}{2}\)= 0

Multiply by two: x+30y+15=0

From our first equation, y=(150/x), and substituting this into our above equation:

x+30(150/x)+15= x+(4500/x)+15=0

Multiply by x: x²+15x+4500=0

Factorize to get (x+60)(x−75)=0

We get either -60 or 75. Speed must be positive, so the answer is 75. 

3 .

6

4.

216+72sqrt3

Maybe it is \(2^6\) but you probably have to minus the repeats.

Which question do you need the help on most?

We aren't here to do all your AoPS question, but it's fine if you want to ask one.

Ur welcome!

I do AOPS I am robot317

I sometimes ask alcumus problems

I do (I am rabbit317)