Example 11-6. How many three digit even numbers can be formed using the
digits { 1 , 2 , 3 , 5 , 7 }if repetition of any digits is not allowed?
Solution A three digit number has the representation (hundreds place)(tens place)(units place).
If the three digit number is to be an even number, then the units place must be filled
with the number 2. Hence there are = 1 ways to perform this task. The tens place
can be filled with any of the numbers 1,3,5,7 and so m= 4 ways to perform this task.
Finally, if one of the numbers 1 , 3 , 5 , 7 is selected for the tens place and there is to be
no repetition of numbers, then only 3 numbers are left for the hundreds place. This
gives n= 3 ways for the hundreds place. This shows that there are n·m·= 3 · 4 ·1 = 12
ways to complete the task.
Each arrangement in an ordered set of items is called a permutation of the set of
items. For example, how many ways can you arrange three books on a shelf? There
are 3 choices for the first book, 2 choices for the second book and 1 selection for the
last book. This gives 3 · 2 ·1 = 3! = 6 ways to arrange three books on a shelf. If the
books are labeled a,b and c, then the 6 arrangements are
abc
bac
cab
acb
bca
cba
In general, the number of permutations of nthings is n-factorial, written n!. That
is, there are nchoices for the first item, (n−1) choices for the second item, (n−2)
choices for the third item, etc. This gives the number of permutations as
nPn=n(n−1)(n−2) ···(3)(2)(1) = n! (11 .31)
The grouping of a selection of mitems from a collection of nitems, m < n , is
called the number of permutations of nitems taken mat a time. For example, to
determine the number of permutations of the letters a,b,c,d taken two at a time,
note that there are 4 choices for the first letter leaving three choices for the second
letter. This gives 12 such arrangements. These arrangements are
ab
ac
ad
ba
bc
bd
ca
cb
cd
da
db
dc
