According to my book:
1.The fundamental principle of counting is used to count the number of possible ways in which a task can be done without actually counting manually.
2.Under the fundamental principle of counting comes the principle of addition and the principle of multiplication.
3.The principle of addition states if a one task can be one done in $m$ ways and another task which is MUTUALLY EXCLUSIVE of the first task can be done in $n$ ways, then the number of possible ways in which either can be done is $m+n$.This result can be generalized.
4.The principle of multiplication states that if one task can be done in $m$ ways and another task which is INDEPENDENT of the first task can be done in $n$ ways, after the first task has been performed, then the number of possible ways in which both the tasks can be done is $m \times n$.This result can be generalized.
5.Permutations refer to the possible arrangements of a collection of objects.
6.Combinations refer to the possible selections of a collection of objects.
My problems:
1.I was having a lot of problems understanding the difference between the principle of addition and the principle of multiplication. I thought about it a lot and this is my interpretation:
(a).The addition principle is applied when we want to calculate the number of possible ways to perform a task (perform any one of the subtasks).
The task consists of $k$ subtasks ($n_1, n_2, n_3, ..., n_k$), the subtasks can be done in $m_1, m_2, m_3, ..., m_k$ ways respectively. Thus the number of possible ways to do the task is $m_1 + m_2 + m_3 +...+ m_k$.
(b).The multiplication principle is applied when we want to calculate the number of possible ways to perform a task(perform all the subtasks).The task consists of $k$ subtasks($n_1, n_2, n_3, ..., n_k$), the subtasks can be done in $m_1, m_2, m_3, ..., m_k$ ways respectively.Thus the number of possible ways to do the task is $m_1 \times m_2 \times m_3 \times ... \times m_k$.
Am I correct?
2.I cannot understand the significance of the word MUTUALLY EXCLUSIVE in point number 3 and the word INDEPENDENT in point number 4. What does it mean when we say that one task is independent of the other and what is its significance here.
3.What is the difference between principle of multiplication and permutation. I guess permutation is a special case of the principle of multiplication.
4.What is the difference between principle of addition and combination.