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Is this  valid story problem? Because I would need a number of questions to go off of 

Thank you so much!

Are you just making stuff up or? 

yes you are correct

Oh Hectictar ... I did not realize you were so clever ..

BUT  are you really sure this is the correct answer.  ???

I only just saw that SmartMathsMan has already recognised your's, and guest's brilliance.

Isn't it nice to be recognised!!  

Hi Old timer,

This is not a proof, ... I just wanted to convince myself that your statement was true.

I considered Pascals triangle which gives the coefficients nCr

Just considering how the pattern works it can be deduced that

ⁿC_r+ⁿC_r+1 = ⁿ⁺¹C_r+1       and      ⁿC_r+1+ⁿC_r+2 = ⁿ⁺¹C_r+2

_and

ⁿ⁺¹C_r+1   +    ⁿ⁺¹C_{r+2  =}    ⁿ⁺²C_r+2

_so

ⁿC_r+ⁿC_r+1   +    ⁿC_r+1+ⁿC_r+2   =   ⁿ⁺²C_r+2

_hence

ⁿC_r+   2ⁿC_r+1  +ⁿC_r+2   =   ⁿ⁺²C_r+2

To prove it I would probably use the fact that    nCr =    n! /(r! * (n-r)! )

But I will admit that I have not worked through this proof.

The number that brought only yoga mats = 36 - 20  =  16

The number that brought only water bottles = 26 - 20  = 6

So......let the number that had neither  =  N

And we have that

42  = 16 + 20 + 6  + N  simplify

42  =  42  + N

So.....N  =  0.......[ nobody forgot both items ]

Using hectictar's diagram..draw altitude BF ......I believe that we have this  :

Angle  ABF  =  45°...so angle BAF  = 45°

And  (1/2)AC  =  2√3

So...using symmetry.....angle DFA  60°

So  angle FDA  =  75°

So.....using the Law of Sines, we can find DF  as follows :

(1/2)AC / sin (75)  =  DF / sin (45)

2√3 / sin (75) = DF / sin (45)

2√3 * sin 45 / sin 75  = DF

2√3 *  √2 / 2 / sin (75)  = DF

√6 / sin (75)  = DF  =  

√6 /  [ ( 1 + √3 ) / 2√2 ]  =

2√12 / [ 1 + √3 ]  =

4√3  ( 1 - √3)  / -2  =

2√3 ( √3 - 1)  =

6 - 2√3  =  DF  ≈ 2.5358

And since DEF is equilateral,   the perimeter is 3 times this  =

18 - 6√3    units

Here's a pic  :

The second question is analogous to the number of ways of putting 4 distinguishable balls into 4 distinguishable boxes with no restrictions  [ that is....a box may contain 0 - 4 balls ]

The number of ways for this to happen is given by : 

n^k     =  4⁴  =    256 ways  

Where  k = the balls  [ the four bracelets ]   and  n = the boxes [ the four people]

First one

f(x)  =  2^x  +  1     g(x)  =  -x + 4

If x  =  1.....then  y  =  3     so   (1,3)  is a solution

Here' the graph  :  https://www.desmos.com/calculator/kn7b69wttw  

Julius has answered the last one.....see his graph....

Sin θ   =  y / r    where  r  =

√  [  (-3)^2  + 7^2 ]   =  √ [ 9 + 49 ]   =  √ 58

So....

sin θ  =  y / r  =    7 / √58

2 dozen  =  24

20   did not contain nuts

18  did not contain chocolate chips

12  did not contain chocolate

 8   did not contain raisins

If eight did not contain rasins, then it is also possible that these eight  also did not contain the other  three ingredients

So.....8  is the largest possible number that did not contain all four ingredients

Number of words that can be made with one vowel and one consonant  =

[ Choose one vowel and one consonant.....and the reverse order of this  ]   =

6 * 20  *  2   =    240

Number of words that can be made with the same vowel repeated = 6

Number of words that can be made with two dissimilar vowels  =

Permute any 2 of the vowels out of 6 =

 P(6,2)  = 30

So.....the total number of possible "words"  is

240 + 6 +  30   =

276 

arc length / circumference   =   central angle in radians / ( 2π )

arc length / ( 2π * 63 )   =   ( 2π/9 ) / ( 2π )

arc length   =   ( 2π/9 ) / ( 2π )  *  ( 2π * 63 )

arc length   =   2π/9  *  63

arc length   =   14π    centimeters

(a)

So

our result   =   5(2 - 8) + \(\frac12\)(12 * 8 - 24)   =   -30 + 36   =   6

(b)

Notice that no matter what number we chose to begin with,

we can calculate our result with this equation:

result   =   5(2 - x) + \(\frac12\)(12x - 24)    , where  x  is the number we chose.

                                                         Let's simplify this and solve for  x .

result   =   10 - 5x + 6x - 12

result   =   x - 2

                               Add  2  to both sides of the equation.

result + 2   =  x

                               In other words...

the number we chose   =  our result + 2  .

To find the number we chose, MathWizard adds  2  to our result.

Let the three angles be    a1,  a1 + d  and  a1 + 2d

So....we have that

a1  +  (a1 + d)  +  (a1 + 2d)  =  180°

3a1  + 3d  =  180°          divide through by 3

(a1  +  d)   =  60°

So......one of the angles must be 60°

Precisely!

Let's use the exponent rules to simplify this expression.

\(\left(2x^2\right)^{-4}=2^{-4}\left(x^2\right)^{-4}\) by the power of a product exponent rule.

\(2^{-4}\left(x^2\right)^{-4}=2^{-4}x^{-8}\)   by the power of a power exponent rule.

\(2^{-4}x^{-8}=\frac{1}{2^4x^8}\)               by the negative power exponent rule.

\(\frac{1}{2^4x^8}=\frac{1}{16x^8}\)                     This step does not involve an exponent rule but rather this is just simplification.

BCH would be 140? :) I imagine you find that by subtracting 40 from both sides, so 180 - 40 = 140

Notice that  ∠BCH  and  ∠HCD  form a straight line, so they must add to  180° .

∠BCH  +  ∠HCD   =   180°

And...

m∠HCD  =  (3x + 13)°        CPhill already found that  x = 9 , so that means

m∠HCD  =  (3(9) + 13)°

m∠HCD  =  (27 + 13)°

m∠HCD  =  40°

So.....remember...

∠BCH  +  ∠HCD   =   180°

∠BCH  +     40°     =   180°

Can you finish it from here? 

They eat cake on July 27, July 28, July 29, July 30, and July 31.  That is  5  days.

They eat cake every day in August.  That is  31  days.

They eat cake on Sept 1, Sept 2, and Sept 3.  That is  3  days.

the number of cake-eating days per year   =   5 + 31 + 3   =   39

They do this in the years  '08 , '09 , '10 , '11 , '12 , '13 , '14 , '15 , and  '16 .  That is  9  years.

39 cake-eating days per year  *  9 years   =   351 cake-eating days

thx i had no idea wow that is truly amazing thank you genious man and God bless you.

Also, 2 + 2 = 4 - 1 = 3.  Quck Mafs 

1 + 1  =   2   

Does this answer your question?