CPhill

The vertex angles  of a regular hexagon  each measure 120°

Therefore,

3m + 40  = 120               2n - 30 = 120      

3m  = 80                         2n =  150

So                                    

6m  = 160

So 

6m + 2n  =

160  + 150  =

310°

(b) I also bought 7 different postcards. How many ways can I send the postcards to my 3 friends, so that each friend gets at least one postcard?

Let k be the number of postcards   and n  be the number of friends

The  number of ways  =

S (k, n) * n!

Where  S(k, n)  is  a  Stirling Number of the Second Kind

So  we have

S(7,3) * 3!  =

301  *  6  =

1806 ways

(a) While travelling abroad, I bought 7 identical bags of candy for 3 friends. How many ways can I distribute the candy to my 3 friends, so that each friend gets at least one bag of candy?

This is similar to  distributing  7 identical balls into  3 distinct boxes with the  restriction  that each  box must contain at least 1 ball

The number of ways is given by

C (7-1, 3 - 1)  =  C ( 6,2)   =  15 ways

√ [53 + 20√ 7]  =   a + b√ c      square both sides

53 + 20√ 7  =  a^2 +2ab√ c  + b^2c

Equating terms

2ab =  20          a^2  +  b^2c   =  53

ab  =10             a^2 + 7b^2  =53

And c  = 7             

Then

a^2 + 7b^2  = 53

Let b =  2

a^2 + 7(2)^2  = 53

a^2 + 28 = 53

a^2  = 25

 a  = 5

So

a + b√ c  =   5 + 2√ 7

So

a + b + c  =   5 +  2 + 7   =  14

Deciphering  the Rosetta Stone  had to be easier than deciphering this.....

EW....I get that  the function will have a range of all real numbers when  a is in the interval  ( -infinity, -1  ]

Yep

Note that the greatest common factor of 60 and 72   =  12

So

12 people get 5 cookies and 6 sticks of gum apiece

144  = 

12 * 12  =

2^2 * 3  * 2^2 * 3

2^4 * 3^2

The greatest power of 2  that is a factor of 144 is 2^4  ⇒  x  = 4  

The greatest power of 3  that is a factor   of   144  is  3^2 ⇒ y  = 2

So

(1/5^{)y - x}   =  (1/5)^{2 - 4}    =  (1/5)^-2   =  5²  =   25

Afer step 1  we have  :  1/n

After step 2 we have :  2/n

After step 3 we have that

n/ 2  = 16       multiply both sides  by 2

n  = 32 =  the starting integer   

\(\frac{31}{11111} \)

equals

0.0027900279(00279)

So....the repeating string is 5 digits in length