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Nine friends sit around a circular table. Two of the friends, April and Dave, want exactly two people to be between them. How many different seatings are possible? (As usual, two seatings are considered the same if one is a rotation of the other.)

Sit the two friends. Two ways.

There are 7 places left 

I believe that the answer is  2*7! = 10080 permutations.

Well.....the circles are tangent.....C  lies on the line  OP....and C is the point of tangency....so....the image seems to be what you describe.......maybe there's something else to the problem  ???

Hi Madyl,

Looks to me like guest has put in a lot of work.

I don't know why you would ask if the answer came from Mathsisfun.  Maybe you have your reasons.

It is good that you have said you do not understand. It is always good to interact with answerers.

Have you spent a lot of time trying to understand?

My criticism is that you have not said thank you (or given a point) for the guests time and attempted explanation.

Do you think guest, or anyone else, should give you more time since you have not shown appreaciation for the time he/she has already spent?

Note:

I have not looked at the question or the answer. I do not know if the answer is correct or if it is well explained.

Note that the total possible arrangements are  5!  =  120

Let's  look at the arrangements where AD do sit together

AD - - -

-AD - -

- - AD -

- - - AD

So  they could occupy  4 different positions....and for each of these they could be seated in 2 ways (AD or DA)

 And for each of these.....the other 3 can be seated in 3! ways

So.....the number of seatingss where they don't  sit together  =

120  - (4)(2)(3!)  =

120   - 8*6  =

120  - 48  =

72

Did you copy this from Math is Fun? I can't really understand this solution. 

PC = 3 

Make sure your question is stated correctly. I do not see anny other answer to that question.

If the answer is PC = 12, than your question is NOT correctly written. 

I think it's not the line OP; it's rather a line that goes through O and P. 

Sudsw....from what you describe, see the image :

PC = 3

*******

We have the form 

P  =  ab^t         where  a = the original population =  27    and t is the number of days

So....we have

36  = 27b^3      divide both sides  by  27

(4/3)  = b^3      take the  3rd root

(4/3)^(1/3)  = b     

So   we want to find this

P  =  27[ *4/3)^(1/3) ] ^30

P =  27 (4/3)^10  =  exact value

P   ≈  479  bees after 30 days

Thanks Chris, I think you overlooked one of the conditions.

I just did this in ful but it was lost.

Anyway here is my pic

\(Area = \frac{135}{360}\times \pi\times 6^2\\ Area = \frac{27\pi }{2} \; units\; squared \)

We can find the radius  of the  circle, R, by using the Law of Cosines

We have that

(9 sqrt (3))^2 =  2R^2  - 2R^2 ( cos 120°)

243  =  2R^2 - 2R^2 * (- 1/2)

243 =  2R^2 + R^2

243  =  3R^2 

Divide both sides by 3

81  = R^2   

Then the area of the  circle =

pi R^2   =

pi *81  =

81 pi  units^2

Q1

4log₃ c + 4log₃ d  =   4 [ log₃ c + log₃ d ]    =   4 log₃ (cd) ]   = log₃ (cd)⁴    TRUE

(3/4) [ Ln a  + Ln b ]  =  (3/4) Ln (ab)  =  Ln(ab)^(3/4)  =  Ln   ( ⁴ √  [a³b³ ] )   FALSE

3 Ln e  + 2 Ln f  =   Ln e³ + Ln f²  =   Ln  (e³ f² )    FALSE

Q2

Note that the points  (0, 1) and (3, 6.4)    are  on the original function

So....the points  (1, 0)  and (6.4,, 3)  are on the one of the  graphs shown

This has to be the graph on the  bottom left

a=7;b=7;c=7;d=7;e=7;f=7;g=7;p=0;cycle:n=a*1000000+b*100000+c*10000+d*1000+e*100+f*10+g;if(n<10000000 and a==g and b==f and c==e, goto loop, goto next); loop:printn," ",;p=p+1; next:g++;if(g<10, goto cycle, 0);g=7;f++;if(f<10, goto cycle, 0);g=7;f=7;e++;if(e<10, goto cycle,0);g=7;f=7;e=7;d++;if(d<10, goto cycle,0);g=7;f=7;e=7;d=7;c++;if(c<10, goto cycle,0);g=7;f=7;e=7;d=7;c=7;b++;if(b<10, goto cycle,0);g=7;f=7;e=7;d=7;c=7;b=7;a++;if(a<10, goto cycle,0);print"Total = ",p

OUTPUT: 7777777  7778777  7779777  7787877  7788877  7789877  7797977  7798977  7799977  7877787  7878787  7879787  7887887  7888887  7889887  7897987  7898987  7899987  7977797  7978797  7979797  7987897  7988897  7989897  7997997  7998997  7999997  8777778  8778778  8779778  8787878  8788878  8789878  8797978  8798978  8799978  8877788  8878788  8879788  8887888  8888888  8889888  8897988  8898988  8899988  8977798  8978798  8979798  8987898  8988898  8989898  8997998  8998998  8999998  9777779  9778779  9779779  9787879  9788879  9789879  9797979  9798979  9799979  9877789  9878789  9879789  9887889  9888889  9889889  9897989  9898989  9899989  9977799  9978799  9979799  9987899  9988899  9989899  9997999  9998999  9999999  Total =  81

P.S. There may be a "bug" in the code !! If so, please let me know. Thanks.

A coin is flipped eight times, and the sequence of heads and tails occurring is recorded. How many distinct sequences are possible?

256    i.e.  2 x 2 x 2 x 2 x 2 x 2 x 2 x 2  also expressed as  2⁸     

_.

There are 13*3 possibilities for the first pair, leaving 12*3 possibilities for the 2nd pair, leaving 11*4 possibilities for the singleton.  So:  13*3*12*3*11*4 = 61776 in total.

There are 7*7*7 = 343 palindromes that can be made.

This solution can be easily  found  graphically : https://www.desmos.com/calculator/8tv2ssgvrv

The area = two triangles each  with a base of  4  and a height of 2

So.....the area  =   2  [ (1/2) (4) (2) ]  =   8 units^2

Yup, this also is incorrect when I enter it into the answer box. Thank you for your time and help though!

This seems to be incorrect. Thank you though!

See the graph  here :    https://www.desmos.com/calculator/qeidqnvtts

The intersection of this line and the circle  will form the diameter of the  circle....to find the radius we can  complete the square  on x  in the equation of the circle......so we have....

x^2 - 12x  +36  + y^2   = 28 + 36

(x + 6)^2  + y^2 =  64

The center is  (6, 0)

The radius  = sqrt (64)   = 8

So....the area we are looking for  is

(1/2) pi (8)^2 =   

32 pi units^2

Um... Just wondering, are there numbers missing from this post?

Happy Belated B-day!!

The circle has a radius of 12, and the line y = 6 - x splits it in half, so the area is 1/2*12^2*pi = 72*pi.