BuilderBoi

Note that for the product to be divisible by 4, there need to be at least 2 even numbers. 

The probability of rolling only odd numbers is \({5 \over 9}^4 = {1025 \over 6561}\)

The probability of rolling exactly 1 even number is \(4 \times 5^3 \times {4 \choose 3} = 2000\)

So, the probability is \(1 - ({1025 \over 6561} + {2000 \over 6561}) = \text{_____}\)

Think about a ratio between the distance Becky travels to the distance Caitlin travels. 

Now, if Becky is 24, how many miles does Caitlin have left to go?

There are \({5 \choose 1} + {5 \choose 2} = 15\) ways to choose 1 or 2 slices of bread. 

There are \(2^7 - 1 = 127\) ways to choose the ingredients (think of it as a binary number 0000000 to 1111111, where each digit represents an ingredient, 0 means it's not there, and 1 means it's there)

So, there are \(15 \times 127 = \text{___}\) sandwichs. 

Note that we use stars and bars for this method to prevent overcounting. 

We have 4 stars and 3 bars to sort the 4 remaining flowers, and the number of orderings is how many ways we can order the stars and bars, making for \({7 \choose 3} = \color{brown}\boxed{35}\) options. 

This is what a path would look like: 

We have the equation: \(x - {1 \over x} = 3\)

Solve as follows: 

\({x^2 \over x} - {1 \over x} = 3\)

\({x^2 - 1 \over x} = 3\)

\(x^2 - 1 = 3x \) 

\(x^2 - 3x - 1 = 0\)

Now, just solve the quadratic...