Probability
# Permutations

How many ways can the letters of the word BOTTLES be arranged such that both of the vowels are at the end?

**Details and assumptions**

The vowels in the word BOTTLES are O and E.

Among $5$ girls in a group, exactly two of them are wearing red shirts. How many ways are there to seat all $5$ girls in a row such that the two girls wearing red shirts are **not** sitting adjacent to each other?

Hint: Treat the two girls as one person. This will help find the number of arrangements that have the girls seated **together**, then subtract the number from 5!, the total number of arrangements.

$10$ people including $A, B$ and $C$ are waiting in a line. How many distinct line-ups are there such that $A, B,$ and $C$ are **not** all adjacent?

**Details and assumptions**

$A, B$ and $C$ may be in any order as long as all three are not adjacent.

3 boys and 2 girls are about to be seated at a round table. If the 2 girls want to sit next to each other, find the number of ways seating these boys and girls.

(Note: since the table is round, we are considering two seating arrangements to be equivalent if they can match just by rotating.)

Mary has enrolled in $6$ courses: Chemistry, Physics, Math, English, French and Biology. She has one textbook for each course and wants to place them on a shelf. How many ways can she arrange the textbooks so that the English textbook is placed at any position to the left of the French textbook?

Hint: There are just as many permutations where the English textbook is to the left of the French textbook as there are permutation where the French textbook is to the left of the English textbook.