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The answer is

\[\binom{38}{5} + \binom{38}{32} = \binom{40}{5}.\]

Finding the Slope (m):

The slope (m) of the line passing through points (-4, 2) and (1, -1) can be found using the following formula:

m = (y2 - y1) / (x2 - x1)

Here, (x1, y1) = (-4, 2) and (x2, y2) = (1, -1)

m = (-1 - 2) / (1 - (-4)) = -3 / 5

Finding the y-intercept (b):

We can use the point-slope form of the equation to find the y-intercept (b). Here, we can use either of the given points. Let's use (-4, 2).

y = mx + b

2 = (-3/5) * (-4) + b

2 = 12/5 + b

b = 2 - 12/5 = -2/5

Writing the Equation in Ax + By = C form:

Now, we can express the equation in the desired form:

A(x) + B(y) = C

Substitute the slope (m) and y-intercept (b):

A(x) + (-3/5)(y) = -2/5

Finding A, B, and C with GCD = 1 and A positive:

To ensure the greatest common divisor (GCD) of A, B, and C is 1 and A is positive, we can multiply the entire equation by the least common multiple (LCM) of the denominators of m and b (which is 5 in this case).

5A(x) - 3y = -2

Since A is positive, we can simply set A = 5:

5(x) - 3y = -2

Therefore, the equation of the line in the desired form is:

5x - 3y = -2

We can write the equation as

\[
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
2 & 3 & 4 & 5 & 1 \\
3 & 4 & 5 & 1 & 2 \\
4 & 5 & 1 & 2 & 3 \\
5 & 1 & 2 & 3 & 4
\end{pmatrix}
\begin{pmatrix}
a \\ b \\ c \\ d \\ e
\end{pmatrix}
=
\begin{pmatrix}
47 \\ 15 \\ 3 \\ 26 \\ 29
\end{pmatrix}
.
\]
 

Inverting the matrix, we get the solution (a,b,c,d,e) = (-5, 4, 14, 3, -2).

The sum of the terms 1/(sqrt(a_1) + sqrt(a_2)) + ... is equal to 45.

The largest possible value of |a_1 - a_6| is 100.

How many ways can you write two different letters chosen from  
A B C D E F G H I J K L M N O P  
so that they are in alphabetical order, and the first letter is a vowel?  

    34 ways  

If A is the first letter, then 15 possibilities    

If E is the first letter, then 11 possibilities    

If  I  is the first letter, then   7 possibilities    

If O is the first letter, then   1 possibility     

Add them up.  

How many miles did I drive?

Ich fuhr mit einer Geschwindigkeit von 40 Meilen pro Stunde zum Strand. Wenn ich stattdessen mit einer Geschwindigkeit von 55 Meilen pro Stunde gefahren wäre, wäre ich 25 Minuten früher angekommen. Wie viele Meilen bin ich gefahren?

\(40\ mph\cdot t=55\ mph(t-25\ min)\\ 40t=55t-1375\ min\\ 15t=1375\ min\\ t=91\frac{2}{3}\ min\)

\(s=v\cdot t=40\ mph\cdot 91\frac{2}{3}\ min\cdot \frac{h}{60\ min}\\ \color{blue}s=61.111\ miles\)

 !

You get the largest D when you take a = -4, b = 4, and c = 4.  This gives you D = 80.

This ended up not being the answer, however, it did say as a hint "Remember that the midpoint of a hypotenuse is a special point." 

We can solve this system of equations by adding and subtracting the equations strategically to eliminate one of the variables.

Adding the Equations:

Adding the two given equations eliminates the term ab^2:

(a^3 + 3ab^2) + (3a^3 - ab^2) = 679 + 615

Combine like terms:

4a^3 = 1294

Solving for a^3:

Divide both sides by 4:

a^3 = 1294 / 4 = 323.5

Subtracting the Equations:

Subtracting the second equation from the first equation eliminates the term 3a^3:

(a^3 + 3ab^2) - (3a^3 - ab^2) = 679 - 615

Combine like terms:

4ab^2 = 64

Solving for ab^2:

Divide both sides by 4:

ab^2 = 16

Relating a and b:

Since we know both a^3 and ab^2, we can try to express one variable in terms of the other.

From the equation for a^3, we can write a =∛(323.5).

Relating a - b:

We want to find a - b. Since we don't have a direct equation for b, we can try to manipulate the equation for ab^2.

Rewrite the equation for ab^2:

b^2 = a(ab^2) = a * 16

Substitute a = ∛(323.5):

b^2 = ∛(323.5) * 16

Now, we can express b in terms of a:

b = ± 4 * ∛(323.5)

Finding a - b:

Since a and b are real numbers, we can consider both positive and negative values of b. However, we only care about the difference a - b.

There are two cases:

Case 1: b = 4 * ∛(323.5)

a - b = ∛(323.5) - (4 * ∛(323.5))

Factor out ∛(323.5):

a - b = ∛(323.5) (1 - 4)

a - b = -3 * ∛(323.5)

Case 2: b = -4 * ∛(323.5)

a - b = ∛(323.5) - (-4 * ∛(323.5))

Factor out ∛(323.5):

a - b = ∛(323.5) (1 + 4)

a - b = 5 * ∛(323.5)

Conclusion:

Since we don't know the signs of a and b beforehand, both cases are valid. Therefore, a - b can be either -3 * ∛(323.5) or 5 * ∛(323.5).

Both answers are negative and positive multiples of the same cube root, so they essentially represent the same value with opposite signs. The absolute value of a - b is:

|a - b| = |(-3) * ∛(323.5)| = |5 * ∛(323.5)| = 5 * ∛(323.5)

We can solve this problem by considering the two unwanted scenarios and subtracting them from the total number of possibilities.

Total number of ways:

There are 3 choices (flavors) for each child, so there are 36=729 total ways for all 6 children to choose their ice cream flavors with no restrictions.

Unwanted Scenario 1: No flavor chosen by exactly 3 children

Choose the flavor that isn't picked by exactly 3: There are 3 ways to choose which flavor will be picked by only 2 or none of the children.

Choose the 3 children who get the non-chosen flavor: There are (36​)=20 ways to choose 3 children out of 6.

Choose the flavors for the remaining 3 children: Each of these children has 2 remaining flavors to choose from, for a total of 23=8 possibilities.

There are 3⋅20⋅8=480​ ways for all flavors to be chosen by either 0, 1, 2, or all 6 children.

Unwanted Scenario 2 (not necessary but can be solved for completeness):

All 3 flavors are chosen by exactly 2 children each. This can be solved similarly to scenario 1, but the calculations are slightly more involved.

Final Answer:

Since we only care about scenarios where exactly one flavor is chosen by 3 children, we subtract the unwanted scenarios from the total:

Total ways - Unwanted Scenario 1 (or Scenario 2) = 729−480 = 249​ ways.

\(Find\ the\ measure\ of\ \angle MON\ in\ degrees.\)

\(\angle CAO+\angle CBO=45°\ |\ peripheral\ \angle \ to\ 90°\ (\angle AOB)\\ \angle MON=90°-\angle CAO+90°-\angle CBO\\ =180°-(\angle CAO+\angle CBO)=180°-45°\\ \color{blue}\angle MON=135° \)

  !

Find AE.

\(\triangle DEF\ is\ equilateral. \\ DA = 3,\ AF = 2,\ EF=5\\ \angle AFE=60°\)

Pythagorean trigonometric theorem: \(a²=b²+c²-2bc\cdot cos\ \alpha \)

\(\overline{AE}\ ^2=\overline{AF}\ ^2+\overline{EF}\ ^2-cos\ (AFE)\\\overline{AE}\ ^2=2^2+5^2-2\cdot 5\cdot cos\ 60^°\\ \overline{AE}= \sqrt{29-2\cdot 5\cdot0.5}\\ \color{blue }\overline{AE}= \sqrt{24}=4.899\)

  !

Find the area of triangle ABC.

\(Point\ X\ is\ on\ side\ BC.\\ AX=13,\ BX=10,\ CX=10\\ \triangle XAB\ symmetrical\ \triangle XAC \\ \color{blue}A_{ABC}= \dfrac{1}{2}\cdot (10+10)\cdot 13=130\)

\(7x^2 + x - 5 = 6x^2 - 11x - 2\) is equal to:
\(x^2+12x-3\)

\((a - 4) + (b - 4)\)

is \(a+b-8\)

sum of roots: \(\frac{-b}{a}\)

sum: \(\frac{-12}{1}\)= -12

-12 - 8 = -20