CPhill

Color any face red

We have a choice of any of the remaining five faces to be red

But  only the oppsite face painted red will  not be adjacent

So....

P ( two adjacent red faces) =  4/5

25  liked one or the other - or both

N ( A or C) =  N(A)  + N (C)  -  N (A and C)

25 = 18 + 12  - N(A and C)

25  = 30 - N(A and C)

N(A and C) = 30 -25

N(A and C) =  5

For N people.....the number of handshakes = 

(N) ( N-1)  / 2

So

(5) (5 - 1)   /2   =

( 5 * 4)  / 2  =

10 handshakes

We can choose  4 of 15 members  and permute each set of these in 4! ways

P (15 , 4) *4!  =  

32760 ways

We have 4 ways to choose the last digit  and  (8)^4  ways for the other digits

8^4 * 4  = 

16384 counting numbers

The original area = s^2

The new area is   (s) (1 - .40) *   (s ) (1 + .60) =

s^2  ( .60) (1.60)  =

.96s^2

The area decreases by    1 - .96 =  .04 =   4%

ADC = 90°

ACD =  60°

AD =  sqrt (3)  times longer than  CD  =     3sqrt (3)

Area of   ACD =   (1/2) (AD) (CD)  = (1/2) (3sqrt (3) ) * (3)  =    (9/2) sqrt (3)  cm^2

Pythagorean Theorem

DG  = sqrt  [ DO^2 + OG^2 ]  = 

sqrt [ 24^2 + 12^2 ] =   

sqrt [ 12^2 ( 2^2 + 1) ] =

12 sqrt  [ 5 ] 

(N + 1)        (N + 2)       (N + 3)                     (N + 19)      (N + 20)

______ *   _______ * _______  *   ......  *    ________ * _______

     N             (N + 1)      (N + 2)                     ( N + 18)     (N + 19)

Note  that the product of all the  fractions will be

(N + 20) / N  =  3

N + 20  = 3N

20 =  2N

N = 10

The first fraction  was   (N + 1)  / N   =       11 /  10

Divide both sides by  13^5

13^17 / ( 13^11 *  13^5)  =    ____

13^17  /( 13^16)  =  ____

13^1  =     13  =    ____