CubeyThePenguin

We can draw a Venn diagram.

office keys || both || wallet

       20       ||    5   ||    30

The probability that you forget your office keys or your wallet is 20 + 30 - 5 = 45%.

So, the answer is 1 - 45% = 55%

The value of \(x\) is \(\boxed{\log_2{7}}\)

Let x be the number of one dollar bills and y be the number of five dollar bills. We can write a system of equations:

14 bills in total ==> x + y = 14                     . (1)

value of bills is $30 ==> x + 5y = 30           . (2)

Multiply equation (1) by 5.

5x + 5y = 70

x + 5y = 30

We get 4x = 40, so x = 10 and y = 4. Sharon has 10 $1 bills and 4 $5 bills.

Once again, we can use Vieta's equations. 

The sum of the roots is \(-\frac{b}{a} = -\frac{-10}{2} = 5\).

The product of the roots is \(\frac{c}{a} = \frac{5}{2}\).

Now, we can simplify: \((2a-3)(4b-6) = 8ab - 12(a+b) + 18 = 8(\frac{5}{2}) - 12(5) + 18 = \boxed{-22}\)

There is probably a formula for this, but I don't know any other method other than brute force.

\((8/5)^2 + (12/5)^2 + (16/5)^2 + \dots + (44/5)^2\)

\((64/25) + (144/25) + \dots + (1936/25)\)

The sum from 64 to 1936 is \((1^2 + 2^2 + \dots 45^2) - (1^2 + 2^2 \dots + 7^2) = \frac{(45)(46)(91)}{6} - 140 = 31225\).

Now, we have \(\frac{31225}{25} = \boxed{1249}\)

The ball hits the ground when the height is 0, so we are solving the equation \(-5t^2+22t+15=0.\)

Using the quadratic equation, we get \(t= -\frac{3}{5}, 5\). Time can't be negative, so the ball hits the ground after 5 seconds.

I am unsure about the second part, but I hope this gets you started.

Let x be the number of cupcakes and y be the number of cookies sold. We can write a system of equations:

total of 250 cupcakes and cookies ==> x + y = 250           . (1)

total cost was $506 ==> 3.5x + 1.25y = 506                       . (2)

We can multiply equation (1) by 5 and equation (2) by -4:

5x + 5y = 1250

-14x - 5y = -2024

Adding, we get -9x = -774, so x = 86 and y = 164.

You baked 86 cupcakes and 164 cookies.

The cost to rent each chair is c and the cost to rent each table is t. We can write the system of equations

cost to rent 6 chairs and 2 tables ==> 6c + 2t = 24       . (1)

cost to rent 3 chairs and 5 tables ==> 3c + 5t = 45       . (2)

Multiply equation (1) by 5 and equation (2) by -2:

30c + 10t = 120

-6c - 10t = -90

So, 24c = 30 and c = 1.25. Therefore, t = 8.25.

We can use Vieta's Formulas. The sum of the roots is \(-\frac{b}{a} = \frac{m}{2}\) and the product of the roots is \(\frac{c}{a} = \frac{n}{2}\).

The sum is 6, so \(\frac{m}{2}=6\) and m = 12.

The product is 10, so \(\frac{n}{a} = 10\) and n = 20.

So, \(m+n=12+20 = \boxed{32}\)

If we can repeat letters more than once, there are \(4^4 = \boxed{256}\) words that can be made.