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52 P 5  =  52! / 47! = 311 875 200 ways to pick 5 cards                ( 47  is   52 - 5 ) 

= x^2 -x -1 = 0     Quadratic Formula   with    a = 1     b = -1      c = -1 

         to find the x values are    1/2 ± sqrt(5) / 2

Like this (assuming my simplification of the numbers is correct):

I'll leave you to solve the above to find the actual value of Cfixed.

Only in Q I   nad Q III   is tangent a positive number      ( remember tan = sin/cos) 

    since the question asks for Φ between  0 and 180 degrees, you only get one answer in Q I

        using a calculator         arctan (9) = 83.7 degrees 

Thanks for your answer, but as Alan has informed us, I think the simplified equation should be \(10f=5m\).

Otherwise, thanks a lot!!

There are 5 vowels: a, e, i, o, and u. There are 4 possible first digits: 1, 2, 4, and 6. For each of the three letters, there are 23 possible letters that can be used (since no letter can be repeated). For each of the three digits, there are 9 possible digits that can be used (since no digit can be repeated).

Therefore, the total number of possible license plates is 5 * 4 * 23 * 23 * 23 * 9 * 9 * 9 = 3000.

There are two cases to consider:

Case 1: Two children sit next to one adult, and one child sits between two adults.

To seat the three adults, there are 2! ways, since rotations of the same arrangement are considered the same.

To seat the three children, there are two possible arrangements:

The child sitting between the two adults can be chosen in 3 ways. The other two children can then be seated in 2! ways.

The two children sitting next to the same adult can be chosen in 3 ways. The other child can then be seated in 2 ways.

Therefore, the total number of ways to seat the adults and children in Case 1 is 2! * (3 * 2! + 3 * 2) = 24.

Case 2: One child sits next to each adult.

To seat the adults, there are 2! ways.

To seat the three children, there are 3! ways.

Therefore, the total number of ways to seat the adults and children in Case 2 is 2! * 3! = 12.

Therefore, the total number of ways to seat the three adults and three children at the circular table if each child must be next to two adults is 24 + 12 = 36.

Total area = pi r^2 + (2r * 2r)      ( rememeber the sides of the square = 2r )

The area =  pi  r^2  +  (2r)^2   =  (pi +4) r^2  =  7.142 r^2

    Then  solving for 'r'  =====>   sqrt (area/7.142) = r

                  First one:  area = 9

                                      r = sqrt ( 9 / 7.142) = 1.1  in

                                         square side = 2r = 2.2 in         ( rounded answers)

                 second one:  area 20 in^2

                               r = sqrt ( 20 / 7.142) =.......     etc

I'll let you do the others...

Rate of SLOWER crane =  1 ship /   (x -10 )

Rate of FASTER crane =   1 ship / x 

Now for one ship with both working the time = 12 hrs:

    1 ship /  ( 1/(x-10)  +   1/x)  =  12  hr       <====     solve for 'x' 

1 = 12/(x-10)  + 12 / x

x^2 - 10x = 12x + 12x-120

x^2 -34x +120 = 0      Use Quadrtaic Formula  ( or factoring) to find x =   30     or 4 (extraneous)

                                         so one crane takes  30 hours      the other  takes  20  hours alone     

Hmm!  Check the left-hand side of 15f = 5m

Let f be the number of female members and m be the number of male members. We know the following:

40f + 25m = 30(f + m)

Simplifying this equation, we get:

15f = 5m

Therefore, the ratio of female to male members is f/m=1/3.

There are always 3 ways to get to the next letter from any given letter: down-left, down, and down-right.

There are 3 transitions: A-R, R-C, and C-H

\(3\times3\times3=\boxed{27}\)

As follows:

There are 2 ways to start with the letter A: either from the first A in the array, or from the second A. To reach the second A, we must move down once. Once we have reached the first or second A, we can move in any direction to reach the letter R. There are 3 ways to do this: we can move down, right, or down and then right. Once we have reached the letter R, we can again move in any direction to reach the letter C. There are 3 ways to do this: we can move down, left, or left and then down. Once we have reached the letter C, the last step is to move up to reach the letter H, and there is only 1 way to do this. Therefore there are 2×3×3×1=18​ ways to spell the word "ARCH" in the array provided.

If there are 21 yellow tiles, then there are 21 tiles along one diagonal. Since the tiles are square, the total number of tiles along one side of the square is 21 + 1 = 22. Therefore, the total number of tiles in the square is 22 * 22 = 484. Since there are 21 yellow tiles, there are 484 - 21 = 463 gray tiles. So the answer is 463

There are 3 boxes, so there are 3 ways to place the first ball. There are 2 boxes left, so there are 2 ways to place the second ball. There is 1 box left, so there is 1 way to place the third ball. There is 1 box left, so there is 1 way to place the fourth ball. Therefore, there are 3 * 2 * 1 * 1 = 6 ways to place the 4 balls in 3 boxes.

so sorry I meant that angle ABC=135 degrees, could you please try that again? But thank you anyway for answering fully and trying!

Of the statements provided, only 4. The line ℓℓ passes through a midpoint of one side of Δ.Δ. must be true.

We can see this by drawing a diagram of the situation:

[asy] unitsize(0.8 cm);

pair A, B, C, D;

A = (0,0); B = (10,0); C = intersectionpoint(arc(A,10,0,180), arc(B,10,180,0)); D = (B + C)/2;

draw(A--B--C--cycle); draw(A--D);

label("A", A, SW); label("B", B, SE); label("C", C, NE); label("D", D, S); [/asy]

If line ℓ divides triangle △ABC into two congruent triangles, then it must pass through the midpoint of one of the sides of the triangle. In this case, line ℓ passes through the midpoint of side BC.

The other statements are not necessarily true. For example, triangle △ABC does not necessarily have two equal sides or two equal angles. Also, line ℓ does not necessarily have to be perpendicular to a side of the triangle.

Here are some examples:

If triangle △ABC is an isosceles triangle, then it will have two equal sides, but line ℓ may not pass through the midpoint of one of the sides.

If triangle △ABC is an equilateral triangle, then it will have two equal sides and two equal angles, but line ℓ may not pass through the midpoint of one of the sides.

If triangle △ABC is a right triangle, then line ℓ may be perpendicular to a side of the triangle, but it may not pass through the midpoint of one of the sides.

Therefore, the only statement that must be true is that line ℓ passes through the midpoint of one side of the triangle.

To find the area of triangle ABC, we can use the following formula:

Area of Triangle = 1/2 * base * height

We know that the base of the triangle is 8 cm (BC), but we need to find the height.

We can use the law of sines to find the height, since we know the length of one side and the measure of one angle.

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

We know that BC = 8 cm and angle ABC = 35 degrees. We want to find the height, which is the perpendicular distance from point A to line BC. This means that the angle between the height and side BC is 90 degrees (since a perpendicular line forms a right angle).

Therefore, we can use the following equation to find the height:

height = AB * sin(ABC) / sin(90 degrees)

height = 6 cm * sin(35 degrees) / sin(90 degrees)

height = 3.38 cm

Now that we know the base and height of the triangle, we can find the area using the formula above:

Area of Triangle = 1/2 * base * height

Area of Triangle = 1/2 * 8 cm * 3.38 cm

Area of Triangle = 13.52 square cm

Therefore, the area of triangle ABC is 13.52 square cm.