geno3141

MONDAY

p = number of words that do not start with M

-- there are 5 choices for the first letter, then 5 choices for the second letter, 4 choices for the third letter,

       3 choices for the fourth letter, 2 choices for the fifth letter, and 1 choice for the last letter.

-- p = 5 x 5 x 4 x 3 x 2 x 1 = 600 arrangements

q = number of words that start with M but do not end with Y:

-- there is only one choice for the first letter and there are 4 choices for the last letter, and 4 choices for

       the second letter, 3 choices for the third letter, 2 choices for the fourth letter, and 1 choice for the

       fifth letter

-- q = 1 x 4 x 3 x 2 x 1 x 4 =  96 arrangements

There are 53 different scores that a student can get:  0, 1, 2, 3, ..., 51, 52.

So, it will take 53(1) + 1 students to guarantee that 2 students have the same score.

Likewise, it will take 53(2) + 1 students to guarantee that 3 students have the same score.

And, it will take 53(3) + 1 students to guarantee that 4 students have the same score.

To guarantee that at least 39 students to have the same score:  53(38) + 1  =  2015.

If they are independent events, multiply the probabilities.

The reason that adding is not appropriate:

-- consider this case:

    -- assume that the probability of A not appearing is 0.90 

    -- assume that the probability of B not appearing is 0.80

-- if you added them together, you would get a probability of 1.70, an impossibilit.

The original roll has a diameter of 5 unts   --->   radius = 2.5 units.

It's total area is  pi·2.5²  =  6.25pi units².

However, not all of that is paper; there is a hollow core of diameter 1 unit   --->   radius = 0.5 units.

The area of the core is  pi·0.5²  =  0.25pi units².

Subtracting the area of the core from the total area gives  6.25pi units² - 0.25pi units²  =  6.00pi units².

To get twice the amount of paper:  2 x 6.00pi units²  =  12.00pi units².

However, we can't forget the core, so we need  12.00pi units² + 0.25pi units²  =  12.25pi units.

Finding the square root of 12.25, we get 3.5, so the roll needs a radius of 3.5 units,

which is a diameter of 7 units.

Since the chairs and the table have no arms, the 10 arms belong to 5 people (assuming that each person has 2 arms).

Therefore, the 5 persons also have 10 legs (again assuming that each person has 2 legs).

Since there is a total of 18 legs, 8 legs belong to the table and the chairs.

Since 4 of the legs belong to the one table, the remaining 4 legs belong to 1 chair.

We have 5 persons and 1 chair; it's time to get 4 more chairs.

If n = 3, then n + 2 = 5, and n + 4 = 7; all primes.

Formula for the n^th term of an arithmetic progression:  t_n  =  t₁ + (n - 1)d

--->   389  =  t₁ + (30 - 1)d   --->   389  =  t₁ + 29d

--->   181  =  t₁ + (14 - 1)d   --->   181  =  t₁ + 13d

Subtracting:                                  208  =  16d

                                                        13  =  d

To find the first term:     181  =  t₁ + 13d   --->   181  =  t₁ + 13(13)

                                                                          181  =  t₁ + 169

                                                                            12  =  t₁

Now, you know that  t₁ = 12  and  d = 13,

use this formula to find the 25^th term:   t_n  =  t₁ + (n - 1)d

And use this formula to find the sum of the first 25 terms:  Sum  =  n( t₁ + t_n ) / 2

where  n  is the number of terms

          t₁  is the value of the first term

          t_n  is the value of the last term.

I did -- thank you!

a₁  =  1

a₂  =  2¹·1  =  2¹

a₃  =  2²·2  =  2³

a₄  =  2³·2³  =  2⁶

a₅  =  2⁴·2⁶  =  2¹⁰

a₆  =  2⁵·2¹⁰  =  2¹⁵  

The exponents form the triangular numbers:  1, 3, 6, 10, 15, ...

The formula for the n^th triangular number is  n(n + 1)/2

This will give you the exponent upon 2  --  but be careful  -- for a₆, you want the 5^th triangular number (etc).

        Ax - 3y  =  6     and     x = -1     and     y = -3:

A(-1) - 3(-3)  =  6

         -A + 9  =  6

                -A  =  -3

                 A  =  3