geno3141

If there were 5 different letters, there would be 5! ways of arranging them.

However, since there were two copies of the letter X, we need to divide that answer by 2!.

And, since there were two copies of the letter E, we need to divide that answer by 2!.

5! / 2! / 2!  =  120 / 2 / 2  =  30 different arrangements of the letters.

[ sqrt( x + sqrt( x + sqrt( x = ... ]  =  5

Square both sides:

x + [ sqrt( x + sqrt( x + sqrt( x = ... ]   =  25

Since  [ sqrt( x + sqrt( x + sqrt( x = ... ]  =  5:

x + 5  =  25

      x  =  20

101 + 103 + 105 + ... + 995 + 997 + 999

This is an arithmetic sequence with     t₁ = 101     t_n = 999     d = 2

We can use the formula:  t_n  =  t₁ + (n - 1)d  to find the number of terms:

--->     999  =  101 + (n - 1)2     --->     n = 450

We can use the formula:  Sum  =  n(t₁ + t_n)/2  to find the sum of the terms:

--->     Sum  =  450(101 + 999)/2     --->     Sum  =  247 500

If the sum were:  (1 - 2 + 3) + (4 - 5 + 6) + (7 - 8 + 9) + ... + (1999 - 2000 + 2001) + (2002 - 2003 + 2004)

we could do this:       2         +       5        +        8        + ... +             2000               +            2003

and we would have an arithmetic series with:  t₁ = 2     t_n = 2003     d = 3.

We could then use this formula;  t_n  =  t₁ + (n - 1)d  to find the number of terms:

     --->     2003  =  2 + (n - 1)3     --->     2001  =  (n - 1)3     --->     667  =  n - 1     --->   n  =  668  

and use this formula:  Sum  =  n(t₁ + t_n)/2  to find the sum:

     --->     Sum  =  668(2 + 2003)/2     --->     Sum  =  669 670

However, we don't have  2004  in the original problem, so we'll have to subtract that:

     --->     669 670 - 2004  =  667 666

If you take out four socks, at least two must match.

The first three can be of different colors but you can't have four socks that are different from one another.

The number of socks of each color does not affect the answer.

         x(10 - x)  =  40

         10x - x²  =  40

       -x² + 10x  =  40

-x² + 10x - 40  =  0

 x² - 10x + 40  =  0

Quadratic formula with   a = 1   b = -10   c = 40

x  =  [ -(-10) + sqrt( (-10)² - 4(1)(40) ] ( 2(1) ]

   =  [ 10 + sqrt( 100 - 160 ) ] / 2

   =  [ 10 + sqrt( -60 ) ] / 2

   =  [ 10 + 2sqrt(-15) ] / 2

   =  5 + sqrt(15)·i

Other answer:  5 - sqrt(15)·i

Use the distance formula:  d  =  sqrt( (x₂ - x₁)² +(y₂ - y₁)² )

--->   10sqrt(2)  =  sqrt( ( 4 - x )² + ( 7 - 21 )² )

Square both sides:

--->   200  =  (4 - x)² + (-14)²

--->   200  =  (4 - x)² + 196

--->       4  =  (4 - x)² 

Find the square root of both sides:

--->     4 - x  =  2          or          4 - x  =  -2

                x  =  2                             x  =  6

Etc. ...

Let  A₁·A₂·A₃· ...  =  X

Square both sides:

     ( A₁·A₂·A₃· ... )²  =  X²

      (A₁)²·[ (A₂)²·(A₃)²·(A₄)²· ... ]  =  X² 

But:  (A₁) = 3     (A₂)² = A₁     (A₃)² = A₂     (A₄)² = A₃  

So:  (A₁)²·[ (A₂)²·(A₃)²·(A₄)²· ... ]  =  3·[ A₁·A₂·A₃· ... ]  =  X²  

and:  A₁·A₂·A₃· ...  =  X   --->         =  3·[ X ]  =  X²     --->   3  =  X

So, the product is 3.

A formula for the area of a parallelogram is:  Area  =  a · b · sin(C)

For it to apply in this situation, we need the size of angle B:  angle(B)  =  180^o - angle(A)  =  135^o.

Area  =  10 · 14 · sin(135^o)  =  ...

Since the values in the table represent the points in a straight line, let's find the slope:

m  =  [ (-17) - (-14) ] / (p + 2) - (p) ]  =  -3 / 2   --->   m  =  -3/2

Now, let's find the equation of the line by using the point-slope formula:

y - (-5)  /  (-3/2)·(x - 2)   --->   y + 5  =  (-3/2)x + 3   --->   y  =  (-3/2)x - 2

For the point  (13, q):  y  =  (-3/2)x - 2   --->   q  =  (-3/2)(13) - 2   --->  q  =  -21.5

For the point  (p, 14):  y  =  (-3/2)x - 2   --->   14  =  (-3/2)(p) - 2   --->   p  =  -32/3

Left for you to finish ...