parmen

There are 52 cards in a deck, so there are 51 pairs of adjacent cards. Each pair has an equal probability of being Aces or Kings, which is 524​⋅513​=22112​. So, the expected number of pairs of adjacent cards that are both Aces or Kings is:

Expected number = 51 * (\frac{12}{221}) = \boxed{\frac{40}{221}}

The distance between two points on a sphere along the minor arc of a great circle is equal to the central angle between the two points in radians. The central angle between two points on a sphere is equal to the arccosine of the dot product of the unit vectors pointing to the two points.

Therefore, to find the distance from A to B along the minor arc of a great circle, we first need to find the unit vectors pointing to A and B. We can do this by dividing the coordinates of A and B by the radius of the sphere, which is 1.

Let a and b be the unit vectors pointing to A and B, respectively. Then the distance from A to B along the minor arc of a great circle is equal to the central angle between a and b, which is given by:

theta = arccos(\mathbf{a} \cdot \mathbf{b})

We know that the space distance between A and B is 3​. We can use this information to find the dot product of a and b.

\mathbf{a} \cdot \mathbf{b} = \cos(theta) = \frac{\sqrt{3}}{2}

Therefore, the distance from A to B along the minor arc of a great circle is given by:

theta = arccos(\mathbf{a} \cdot \mathbf{b}) = arccos(\frac{\sqrt{3}}{2})

theta = \frac{\pi}{3}

Therefore, the distance from A to B along the minor arc of a great circle is pi/3.

1. If the balls and boxes are both distinguishable, then we can place the balls in the boxes in 84 ways.

2. If neither the balls nor the boxes are distinguishable, then we are simply putting 4 indistinguishable objects into 8 distinguishable boxes. This can be done in (48+4−1​)=(411​) ways.

3. If the balls are indistinguishable, and the boxes are distinguishable, then we can put the balls in the boxes in 8 ways. We can then permute the balls within each box in 4! ways. This gives us a total of 8⋅4!=192 ways.

4. If the balls are distinguishable, and the boxes are indistinguishable, then we can simply put the balls in the boxes in 4! ways.

Therefore, the answers are:

1.84=4096​2.(411​)=330​3.8⋅4!=192​4.4!=24​

Letxbe the fraction of the swim team that are sophomores. Then, the fraction of the swim team that are freshmen is1−x.

We are given that the probability that a sophomore is elected is4/7times the probability that a freshman is elected. This can be expressed mathematically as follows:

P(\text{sophomore}) = 4/7 \cdot P(\text{freshman})

We are also given that each member is equally likely to be elected, which means that the probability of a sophomore being elected is equal to the probability of a freshman being elected plus the probability of a sophomore being elected. This can be expressed mathematically as follows:

P(\text{sophomore}) = P(\text{freshman}) + P(\text{sophomore})

Substituting the first equation into the second equation, we get:

4/7 \cdot P(\text{freshman}) = P(\text{freshman}) + P(\text{sophomore})

Combining like terms, we get:

3/7 \cdot P(\text{sophomore}) = P(\text{freshman})

Dividing both sides byP(freshman), we get:

3/7 \cdot x = 1-x

Solving forx, we get:

x = \frac{7}{10}

Therefore, 7/10 ​​of the swim team are sophomores.

To computecos72 p.m​, we can use the cosine addition formula: \begin{align*} \cos (x + y) &= \cos x \cos y - \sin x \sin y\ &= \cos x \cos y + \sqrt{1 - \cos^2 x} \sqrt{1 - \cos^2 y}, \end{align*}where xandyare any angles.

We know thatcos7Pi​=748​​andcos73 p.m​=721​​,so we can plug these values into the cosine addition formula to get: \begin{align*} \cos \frac{2 \pi}{7} &= \cos \left( \frac{\pi}{7} + \frac{3 \pi}{7} \right)\ &= \cos \frac{\pi}{7} \cos \frac{3 \pi}{7} - \sin \frac{\pi}{7} \sin \frac{3 \pi}{7}\ &= \frac{\sqrt{48}}{7} \cdot \frac{\sqrt{21}}{7} - \sqrt{1 - \left( \frac{\sqrt{48}}{7} \right)^2} \sqrt{1 - \left( \frac{\sqrt{21}}{7} \right)^2}\ &= \frac{6 \sqrt{3} - \sqrt{21}}{7}. \end{align*}Therefore, cos72 p.m​=763​−21​​​.

By the Pythagorean identity, we have sin2i+cos2i=1. Substitutingcosi=113​,we getsin2i+(113​)2=1,sosin2i=12196​.

Taking the square root of both sides, we get sini=±1196​​. Sinceiis acute,sini must be positive. Therefore, sini=1196​​​.