In general, the number of permutations of nthings taken mat a time is given by
the formula
nPm=n(n−1)(n−2) ···(n−m+ 1) (11 .32)
which is a product of m-factors. In the special case m=n,the number of permutations
of nthings taken all the time is nPn as given by equation (11.31).
If in the collection of items there are n 1 repeats on one item, n 2 repeats of
another item,.. .,nm repeats of still another item, then many of the total number
of permutations will be repeats. To remove these repeats just divide by factorial
associated with the number of repeats. This gives the permutation of nthings taken
all at a time as
P= n!
n 1 !n 2 !···nm!
where n=n 1 +n 2 +n 3 +··· +nm (11 .33)
Combinations
A collection of items without regard to the order of arrangement is called a
combination. For example, abc,bca,cab,bac,cba,acb all represent the same collection
of the letters a,b and c, where the different permutations are ignored. The number
of combination of( nthings taken mat time is denoted by using either of the notations
n
m
)
or nCmand is calculated as follows. If nCmor
(
n
m
)
is a collection of mitems from
a set of nitems, then for each combination of mthings there are m!permutations so
that one can write
nPm=m!
(
n
m
)
=m!nCm
or nCm=
(
n
m
)
=n
Pm
m! =
n(n−1)(n−2) ···(n−m+ 1)
m!
= n!
m!(n−m)!
n≥ 0 , 0 ≤m≤n
(11 .34)
The term
(
n
m
)
represents the mth binomial coefficient.
Note that binomial coefficients
(
n
0
)
= 1 and
(
n
n
)
= 1 and that
(
n
m
)
=
(
n
n−m
)
The binomial coefficients also satisfy the recursive property
(
n
m
)
+
(
n
m+ 1
)
=
(
n+ 1
m+ 1
)
m≥ 0 and integral
