25 books are placed at random in a book shelf.
What is the probability that particular three books shall
be placed
. Always together
. Never together
Always together:
Successful arrangements:
Imagine tying a string around the three particular books, and
considering that package as a single "thing". Then there are
only 23 "things" to arrange, 22 non-particular books and 1
group of the three particular books tied together. There are
3! ways to arrange the 3 particular books before tying them
together, and for each of those, there are 23! ways to arrange
the 23 "things".
The number of successful arrangements is (3!)(23!)
The number of possible arrangements is 25!
Probability = [(3!)(23!)]/(25!) = 1/100
-----------------------
Never together
Successful arrangements:
Think of first putting the 22 non-particular book on
the shelf, and then inserting the three particular books
among them so that no two are together.
For each of the 22! arrangements of the 22 non-particular
books on the shelf there are 23 places to put the three
particular books. To explain, these 23 places are:
1. on the far left, (left of the 1st non-particular book)
2. between the 1st and 2nd non-particular books
3. between the 2nd and 3rd non-particular books
4. between the 3rd and 4th non-particular books
...
...
20. between the 19th and 20th non-particular books
21. between the 20th and 21st non-particular books
22. between the 21st and 22nd non-particular books
23. on the far right, (right of the 22nd non-particular book)
Number the 3 particular books first, second and third.
There are 23 places to put the first particular book.
There are then 22 places to put the second particular book.
There are then 21 places to put the third particular book.
The number of successful arrangements is (22!)(23)(22)(21)
The number of possible arrangements is 25!
Probability = (22!)(23)(22)(21)/(25!) = 77/100
Edwin
