CPhill

(sec x + tanx ) ^2  =

(1/ cosx   + sinx / cosx)^2  =

(1 + sin x)^2          (1+ sin x) (1 + sin x)            (1 + sin x) (1 + sinx)             1 + sin x

_________      =   _________________  =  ___________________ =   ___________

    cos^2x                1 -  sin^2x                         (1 + sin x)(1 - sin x)              1 -   sin x

4 letter words with 4 consonants  =  Any of 3 consonants can occupy all 4 positions  = (3)^4  = 81 words

4 letter words with 3 consonants  =   Any of  the 3 consonants can be chosen for any 3  of 4 positions and we have 2 choices for the final position of the vowel....so (3)^3 * 4C3 * 2  =  216 words

4 letter words with 2 consonants  =  Any of the 3 consonants can be chosen for 2 of the 4 positions  and we have  2^2 ways to choose the vowels for the other 2 postions   =    (3)^2 * 4C2 * 2^2  =  216 words

4 letter words with 1 consonant =  And of the 3 consonants can be chosen for 1 of the four positions  and we have (2)^3  ways to choose the vowels for the other 3 positions  =  (3)  * 4C1 * 2^3  = 96 words

So....

81 +  216(2) +96  =  609 words

Another way to see this is that we have  (5)^4  =  625  possible words

And  the number of words that contain no consonants  = (2)^4  = 16

So...the number of words with at least one consonant  =  625 - 16  = 609 words

LOL!!!!.......I didn't even notice that   !!!!!

Evaluate   c + 1/a

a + 1/b  =  1             b + 1/c  =  1   

a = 1 - 1/b                   1/c   =   1 - b

a = (b - 1)                    c  =  1 

      _____                          ____

          b                               1 - b

1/a  =    b

          ____

            b - 1

So

c +  1/a  =

   1         +         b

____                ____      =         [factor a negative out of the second denominator  ]

(1 - b)               b - 1

  1        -         b

____          _______     =

1 - b             1 - b

( 1 - b)

______   =

(1  - b)

1

Very nice, heureka.....I Iike this problem  !!!

P(6)   =  (2 on first roll) (4 on second roll) + (4 on first roll)(2 on second roll)   =

               (1/4) (2/3)     + (1/2) (1/3)  =

                   2/12  + 1/6  =

                    1/6  + 1/6  =    

                         1/3

P(7)  = (2 on first roll) (5 on second roll) + (5 on first roll)(2 on second roll)  =

              (1/4) (1/3)    + (1/4) (1/3)  =

                    1/12   + 1/12

                        1/6

P(8)   =  (4 on first roll) (4 on secod roll)  =

                   (1/2) (1/3)  =

                          1/6

P(9)  =  ( 4 on first roll )(5 on second roll) + (5 on first roll)(4 on second roll) =

                (1/2) (1/3)    + (1/4) (2/3)  =

                    1/6   +  2/12  =

                         1/3

So

P(6)  = 1/3

P(7 or 8)  =  1/6 + 1/6  = 1/3

P(9)  = 1/3

"A"   provides a fair division

Yep...sorry....I missed a sign!!!!

In a particular game, a spinner with four equally-sized sectors labeled 1, 4, 6, and 8 is spun twice. One turn is considered 2 spins of the spinner.

If the sum of the spins is even, you move forward 6 spaces. Otherwise, you move back 2 spaces.

What is the mathematical expectation for the number of spaces moved in one turn?

A) 1 space forward

B) 1 space backward

C) 3 spaces forward

D) 3 spaces backward

Possible outcomes

(1,1)      (4,4)     (6, 6)  (8,8)

(1,4)      (4, 6)    (6, 8)

(4, 1)     (6,4)     (8,6)

(1, 6)     (4,8)

(6.1)      (8,4)

(1,8)

(8,1)

P(even )  = 10/16  =  5/8

P(odd)  =  (3/8)

So.....the expectation is

(5/8)(6)  - (3/8)(2)   =  30/8 - 6/8  =   24/8  =  3 spaces forward

1) Forty eight slips of paper are numbered 1 through 48 and are distributed among a group of people. A random number generator is used to select a single number between 1 and 48, inclusively.

A fair decision is made using this process.

How many people could be in this group?

A) 5

B) 8

C) 9

D) 10

The number of people must be a divisor of 48.......so   "B"  is the answer.....

14-(-5x)+2z-(-32)+4z-2x  =

14 + 5x+2z +32 + 4z  - 2x     [ combine like terms ]

[14 + 32]  + [ 5x -2x ] + [ 2z + 4z ]  =

46 + 3x + 6z

EDIT....