How many different linear arrangements are there of the letters a, b,c, d, e for which: (a a is last in line? (b a is before d? (c a, b and c are next to each other?

A) Since a is last in line, we can disregard a, and concentrate on the remaining letters.
Let's start by drawing out a representation:

_ _ _ _ a

Since the other letters don't matter, then the number of ways simply becomes 4! = 24 ways

b) Since a is before d, we need to account for all of the possible cases.

Case 1: a d _ _ _ 
Case 2: a _ d _ _
Case 3: a _ _ d _
Case 4: a _ _ _ d

Let's start with case 1.
Since there are four different arrangements they can make, we also need to account for the remaining 4 letters.


Now, for case 2:
Let's group the three terms together. They can appear in: 3 spaces.


Case 3:
Exactly, the same process. Account for how many times this can happen, and multiply by 4!, since there are no specifics for the remaining letters.






c) Let's start by dealing with the restrictions.
By visually representing it, then we can see some obvious patterns.

a b c _ _

We know that this isn't the only arrangement that they can make.
From the previous question, we know that they can also sit in these positions:

_ a b c _
_ _ a b c

So, we have three possible arrangements. Now, we can say:
a c b _ _ or c a b _ _
and they are together.

In fact, they can swap in 3! ways. Thus, we need to account for these extra 3! and 2! (since the d and e can swap as well).

His very birth shows and anticipates his heroic nature and fate: "when the gods fashioned gilgamesh, to him they gave a perfect form." so, he was envisioned by gods - therefore, he is not an ordinary man. " the glorious sun shamash bestowed upon him glory; adad the terrible god of storms bestowed upon him courage. the great gods perfected his magnificence beyond all others, terrible like the great wild bull. two thirds god they made him; one third man they made him." (quotations from the tablet 1.) from the fact that he was made by gods, we can tell that he has an important mission ahead of him, something that would be a great contribution to all the humanity. the fact that he is partly human means that he will have to face that most human of all moments - death.
not only was he born to be a hero; he also undertook many heroic acts. one of the most prominent acts was killing the beast humbaba (tablets 2-5), in spite of enkidu's and people's warnings: "[humbaba,]  whose roar is a flood, whose mouth is fire, whose breath is death". gilgamesh does it to earn fame, which is his human trait. but his physical strength is superhuman, so he manages to kill the beast.