Permutations without Repetitions 77
5.4 Permutations without Repetitions
502 Definition We define the symbol factorial, as follows: 0 = 1, and for integer n
≥ 1, n = 1
· 2 · 3···n. n is read n factorial.
503 Example We have
1 =
1 ,
2 =
1 · 2 = 2,
3 =
1 · 2 · 3 = 6,
4 =
1 · 2 · 3 · 4 = 24,
5 =
1 · 2 · 3 · 4 · 5 = 120.
504 Example We have
7 4
= 7
· 6 · 5 · 4 4
= 210 ,
n + 2 n
= n + 2n + 1n
n = n + 2n + 1
, n − 2
n + 1 =
n − 2 n + 1nn − 1n − 2
= 1
n + 1nn − 1 .
505 Definition Let x
1
, x
2
, . . . , x
n
be n distinct objects. A permutation of these objects is simply a rearrangement of them.
506 Example There are 24 permutations of the letters in MAT H, namely
MAT H MAHT
MTAH MT HA
MHTA MHAT
AMT H AMHT
AT MH AT HM
AHT M AHMT
TAMH TAHM
T MAH T MHA
T HMA T HAM
HAT M HAMT
HTAM HT MA
HMTA HMAT
507 Theorem Let x
1
, x
2
, . . . , x
n
be n distinct objects. Then there are n permutations of them.
Proof: The first position can be chosen in n ways, the second object in n − 1 ways, the third in n − 2, etc. This gives
nn − 1n − 2 ···2 · 1 = n.
❑
508 Example The number of permutations of the letters of the word RET ICU LA is 8 = 40320.
78 Chapter 5
509 Example
A bookshelf contains 5 German books, 7 Spanish books and 8 French books. Each book is different from one another.
➊ How many different arrangements can be done of these books?
➋ How many different arrangements can be done of these books if books of each language must be next to each
other? ➌ How many different arrangements can be done of these
books if all the French books must be next to each other?
➍ How many different arrangements can be done of these books if no two French books must be next to each
other? Solution:
➊ We are permuting 5 + 7 + 8 = 20 objects. Thus the number of arrangements sought is
20 = 2432902008176640000. ➋ “Glue” the books by language, this will assure that
books of the same language are together. We permute the 3 languages in 3 ways. We permute the German
books in 5 ways, the Spanish books in 7 ways and the French books in 8 ways. Hence the total number of
ways is 3578 = 146313216000.
➌ Align the German books and the Spanish books first. Putting these 5 + 7 = 12 books creates 12 + 1 = 13
spaces we count the space before the first book, the spaces between books and the space after the last
book. To assure that all the French books are next each other, we “glue” them together and put them in one of
these spaces. Now, the French books can be permuted in 8 ways and the non-French books can be permuted
in 12 ways. Thus the total number of permutations is 13812 = 251073478656000
. ➍ Align the German books and the Spanish books first.
Putting these 5 + 7 = 12 books creates 12 + 1 = 13 spaces we count the space before the first book, the
spaces between books and the space after the last book. To assure that no two French books are next to
each other, we put them into these spaces. The first French book can be put into any of 13 spaces, the
second into any of 12, etc., the eighth French book can be put into any 6 spaces. Now, the non-French books
can be permuted in 12 ways. Thus the total number of permutations is
13121110987612 ,
which is 24856274386944000 .
Homework
510 Problem How many changes can be rung with a peal of five bells?
511 Problem A bookshelf contains 3 Russian novels, 4 German novels, and 5 Spanish
novels. In how many ways may we align them if ➊ there are no constraints as to grouping?
➋ all the Spanish novels must be together? ➌ no two Spanish novels are next to one another?
512 Problem How many permutations of the word IMPURE are there? How many
permutations start with P and end in U? How many permutations are there if the P and the U must always be together in the order PU? How many permutations are there in
which no two vowels I, U, E are adjacent? 513 Problem
How many arrangements can be made of out of the letters of the word
DRAUGHT, the vowels never separated? 514 Problem AIME 1991
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how
many rational numbers between 0 and 1 will 20 be the resulting product?
515 Problem AMC12 2001 A spider has one sock and one shoe for each of its eight
legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?
516 Problem How many trailing 0’s are there when 1000 is multiplied out?
517 Problem In how many ways can 8 people be seated in a row if
➊ there are no constraints as to their seating arrangement? ➋ persons X and Y must sit next to one another?
➌ there are 4 women and 4 men and no 2 men or 2 women can sit next to each other?
➍ there are 4 married couples and each couple must sit together? ➎ there are 4 men and they must sit next to each other?
Permutations with Repetitions 79
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