tertre

Redacted: Check the side lengths.

Try to use Cauchy-Schwarz Inequality because of the sum of squares. 

You can also use this formula. \(|30h-5.5m|\), where \(h\) denotes the number of hours and \(m\) is the number of minutes. Thus, the answer is \(|150-242|=|-92|=92\) degrees.

I'll give an explanation on why it is.

The red marbles, blue marbles, and green marbles that Tom has can be arranged as RRRRRRBBBBGGG, where R denotes the red marbles, B denotes the blue marbles, and G denotes the green marbles.

Tom has, therefore, 6+4+3=13 marbles, so we would assume that if each ball was its own different color, then there would be 13!(Thirteen Factorial) ways to arrange the balls. However, there are six(6) red balls, four(4) blue balls, and three(3) green balls, so these are duplicates of the three colors(red, blue, and green). Thus, the number of ways to arrange the thirteen balls are (13!) / (6! * 4! * 3!), \(\frac{13!}{6!*4!*3!}\)which is \(4*7*3*5*11*13=\boxed{60060}\) distinct ways that Tom can line the thirteen balls up. 

Euler's Theorem gives us 9^2004phi100=9^4  mod 100=3^8 mod 100=6561 mod 100= 61 mod 100, and 6+1=7.

Let's try to break this problem down.

If the average score Laila got was 82, then her total score was 82*5=410 points.

We can label the tests that she took as x, x, x, x, y. The x's are the tests that she got the same score on, and y is the score that she got a different score on.

Now, let's try some casework.

Since she received a higher score on the last test, let's pick a value, and maybe one-hundred(100)? If she has score 100 on the last test, then she must have gotten 4x+100=410, 4x=310, x=77.5 points on the tests she had the same score on. Yet, this does not work, because she scored an integer number of points on all five tests.

 Thus, our goal is when we subtract the value from 410, the result should be divisible by four. If y=98, and if she had scored ninety-eight points on the last test, we have her scoring seventy-eight(78) points on the remaining four tests. This is one possibility. 

If 98 works, then 94 should work, as this also yields a multiple of four. If she has scored 94 points on the last test, then she would have scored 79 points on the remaining tests,

 Trying some of them, y=90 point works, then x=80 points. Also, y=86 points works, then x=81 points. However, if y=82 points, then x=82 points. We do not want this, such as given in the problem. Thus, the only possible values of Laila scoring the number of points on the last test, which is denoted as (y) is, y=98, y=94, y=90, and y=86. Thus, only four(\(\boxed{4}\)) possibilities work.

Hope this helps,

-tertre

We can choose a triangle 8C3 ways or 8!/3!*5!=8*7=56 ways. If the three points are adjacent, then we have eight ways to place the points. and if we have two adjacent points, we have eight*four-=thrity-two possible cases. Thus, the answer is 8+32/56=40/56=5/7.

First, draw a diagram. Label the information that has been presented to you in the problem, then male the steps and moves.

Drawing an altitude from the top of the base to the bottom, we should cleverly use the six units and eight units. By the Pythagorean Theorem, the diagonal length is ten inches. Now, take half of the diagonal, which is five, and point it towards the corner. Here, we should see a 5-12-13 triangle, and thus the height is twelve inches. The volume of the pyramid is \(\frac{B*h}{3}=\frac{48*12}{3}=48*4=\boxed{192}\) squared units(units squared).

We have the Vectors: \(AB=(-12, 12, 0), AB=<-12, 12,0>\) and \(BC=( 0, -12, 6), BC=<0, -12, 6>.\)

Using the cross-product, and the equation of the plane, we get(heft calculations), but this simplifies to \(\frac{12\sqrt{6}}{6}=\boxed{2\sqrt{6}}.\)