1.Algebra Booster

5.4 Algebra Booster
22. ARRANGEMENT INTO GROUPS
(i) The number of ways in which n different things can be
arranged in r different groups
= n + r – 1 P n
, when blank groups are permitted
(ii) The number of ways in which n different things can be
arranged in r different groups is
= n – 1 C r – 1
¥ n!, when blank groups are not permitted.
(iii) The number of ways in which n different things can be
distributed into r different groups
= r n – r C 1
(r – 1) n + r C 2
(r – 2) n – … + (–1) r –1 r C r – 1
,
where blank groups are permitted.
(iv) The number of ways in which n identical things can be
distributed into r different groups is
= n – 1 C r – 1
, where blank groups are not allowed.
(v) The number of ways in which n identical things can be
distributed into r different groups is
= n + r – 1 C r – 1
, where blank groups are allowed.
(vi) If a group has n things in which r are identical, the
number of ways of selecting k things from a group is
Ï
r
n-
r
Ô Â Ck:
k £ r
Ô k = 0
= Ì
k
Ô n-
r
ÔÂ
Ck:
k > r
Ór=
k
23. DE-ARRANGEMENT
Assume x 1
, x 2
, x 3
, …, x n
be n distinct objects such that their
positions are fixed in a row. If we now re-arrange x 1
, x 2
,
x 3
, …, x n
in such a way that no one occupy its original positions,
such an arrangement is called de-arrangement.
For example, a de-arrangement of
1 2 3 4 5 6
is
2 3 5 1 6 4
A de-arrangement of x 1
, x 2
, x 3
, …, x n
is a bijection. Thus
f :{x 1
, x 2
, …, x n
} Æ {x 1
, x 2
, …, x n
} is a bijective function such
that f(x i
) π x i
, i = 1, 2, …, n.
If n things are arranged in a row, the number of ways in
which they can de-rangement in such a way that no one of them
occupied as its original positions is denoted by D(n) and is
Ê 1 1 1 1 n 1ˆ
defined as Dn ( ) = n! ¥ Á1 - + - + - + (-1)
Ë 1! 2! 3! 4! n!
˜
¯
When r things goes to wrong place out of n things, (n – r)
things goes to its original place.
Then D n
= n – r C r
¥ D r
, where D n
be the number of group, if
all n things goes to the wrong place and D r
be the number of
ways, if r things goes to the wrong place.
24. MULTINOMIAL THEOREM
24.1 Combination with Repetitions
(i) The number of combinations of r things out of n things
of which p alike of one kind, q alike of second kind,
s alike of third kind and rest (n – p – q – s) things are
all different
= Co-efficient of x r in
(1 + x +…+x p )(1 + x +…+x q )(1 + x +…+ x s )
(1 + x)(1 + x) … to (n – p – q – s) times
= Co-efficient of x r in
Ê
p+ 1ˆÊ q+ 1ˆÊ s+
1ˆ
1- x 1- x 1-
x
Á (1 + x)
Ë 1- x
˜Á
¯Ë 1- x
˜Á
¯Ë 1-
x
˜
¯
n-p-q-s
(ii) The number of combinations of r things out of n things
of which p alike of one kind, q alike of second kind, s
alike of third kind, when each things is taken at least
once
= Co-efficient of x r in
(x + x 2 +…+ x p )(x + x 2 +…+ x q )(x + x 2 +…+ x s )
= Co-efficient of x (r – p – q – s) in
Ê
p+ 1 q+ 1 s+
1
1- x ˆÊ1- x ˆÊ1-
x ˆ
Á
Ë 1- x
˜Á
¯Ë 1- x
˜Á
¯Ë 1-
x
˜
¯
24.2 Permutations with Repititions
(i) The number of permutations of r things out of n things
in which p alike of one kind, q alike of second kind,
s alike of third kind and so on.
= Co-efficient of x r in
Ï 2 p
2
q
Ô Ê x x x ˆÊ x x x ˆ
Ìr! ¥ Á1 + + + + 1
1! 2! p! ˜Á + + + +
1! 2! q!
˜
ÔÓ
Ë ¯Ë ¯
Ê
2
s
x x x ˆ ¸Ô
Á1 + + + + ˝
Ë 1! 2! s !
˜
¯ Ô˛
(ii) The number of ways in which n identical things can
be distributed into r groups so that no group contains
less than m things and more than k things (m < k) is
Co-efficient of x n in
(x m + x m + 1 + x m + 2 + … + x k ) r
= Co-efficient of x n – mr in
(1 + x + x 2 + … + x k – m ) r
Ê
k
1 - x
= Co-efficient of x n – mr in Á
Ë 1 - x
- m+
1
r
ˆ
˜
¯
¥ (1 - x)
25. SOLUTIONS OF THE EQUATION WITH THE HELP OF
MULTINOMIAL THEOREM
(i) The number of positive integral solutions of the equation
x 1
+ x 2
+ + x m
= n.
Hence, the required number of solutions
= Co-efficient of x n in (x + x 2 + … + x n ) m
= Co-efficient of x n – m in
(1 + x + x 2 + … + x n – 1 ) m
Ê
n
1 - x ˆ
= Co-efficient of x n – m in Á
Ë 1 - x
˜
¯
= Co-efficient of x n – m in (1 – x n ) m ¥ (1 – x) –m
m
-r