MATHalino

Probability
For outcomes that are equally likely to occur:

If the probability of an event to happen is p and the probability for it to fail is q, then

Problem
In how many ways can the letters of the word MATHALINO be arranged if the vowels are to come together?
 

Problem
In how many ways can the letters of the word MATHEMATICS be arranged if the consonants are to come together?
 

Definition
A set is a collection of explicitly-defined distinct elements.
 

Elements of a Set
If 2, 4, 6, 8 are elements of A, we can then write

or it can be written using the set-builder notation

read as "A is the set of all x such that x is an even digit".
 

Fundamental Principle of Counting
If event E₁ can have n₁ different outcomes, event E₂ can have n₂ different outcomes, ..., and event E_m can have n_m different outcomes, then it follows that the number of possible outcomes in which composite events E₁, E₂, ..., E_m can have is

We call this The Multiplication Principle.
 

Problem
In an organization there are CE’s, EE’s and ME’s. The sum of their ages is 2160; the average age is 36; the average age of CE’s and EE’s is 39; the average age of EE’s and ME’s is 32 and 8/11; the average age of the CE’s and ME’s is 36 and 2/3. If each CE had been 1 year older, each EE 6 years and each ME 7 years older, their average age would have been greater by 5 years. Find the number of CE, EE, and ME in the group and their average ages.
 

Situation
A tank is supplied by two pipes A and B and emptied by a third pipe C. If the tank is initially empty and all pipes are opened, the tank can be filled in 20 hours. If the tank is initially full and A and C are opened, the tank can be emptied in 4 hours. If the tank is initially full and B and C are opened, the tank can be emptied in 2 hours. Pipe A supplies 50 liters per minute more than B.
1.   Find the rate of pipe A in liters per minute.

2.   Find the rate of pipe C in liters per minute.

3.   Find the capacity of the tank in liters.

Problem
Find the area individually enclosed by the following Cardioids:
(A)   $r = a(1 - \cos \theta)$
(B)   $r = a(1 + \cos \theta)$
(C)   $r = a(1 - \sin \theta)$
(D)   $r = a(1 + \sin \theta)$
 

Problem
Find the area bounded by $r = a \sin 3\theta$ and $r = a \cos 3\theta$.
 

Problem
Find the area bounded by $r = a \sin 2\theta$ and $r = a \cos 2\theta$.