Multiple separation conditions combined

4 Richard has 3 blue candles, 2 red candles and 6 green candles. The candles are identical apart from their colours. He arranges the 11 candles in a line.

1. Find the number of different arrangements of the 11 candles if there is a red candle at each end.
2. Find the number of different arrangements of the 11 candles if all the blue candles are together and the red candles are not together.

7

1. Find the number of different arrangements of the 9 letters in the word DELIVERED in which the three Es are together and the two Ds are not next to each other.
2. Find the probability that a randomly chosen arrangement of the 9 letters in the word DELIVERED has exactly 4 letters between the two Ds.
   Five letters are selected from the 9 letters in the word DELIVERED.
   [0pt]
3. Find the number of different selections if the 5 letters include at least one D and at least one E . [3]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8. A teacher has 10 different mathematics books. Of these books, 5 are on Algebra, 3 are on Calculus and 2 are on Trigonometry. The teacher arranges all 10 books in random order on a shelf.
a) Find the probability that the Calculus books are next to each other and the Trigonometry books are next to each other. \section*{In this question you must show detailed reasoning.} b) Find the probability that 2 of the Calculus books are next to each other but the third Calculus book is separated from the other 2 by at least 1 other book.