1.Algebra Booster

5.2 Algebra Booster
8. RESTRICTED PERMUTATIONS
String Method
(i) Togetherness
The number of permutations of n different things taken all at
a time when m specified things always come together is given
as (n – m + 1)! ¥ m!.
(ii) Non-togetherness
The number of permutations of n different things taken all at a
time when m specified things never come together is given as
n! – (n – m + 1)! ¥ m!.
9. RANK OF A WORD IN DICTIONARY
Rank of a word is the position of that word, when we arrange
them in alphabetic order according to the English dictionary.
To understand the concept of the rank of a word we should
remember the following steps.
Consider the three letters A, B and C.
The number of arrangement of the letters A, B and C is 3!
and their order is
ABC—1st word
ACB—2nd word
BAC—3rd word
BCA—4th word
CAB—5th word
CBA—6th word
If I ask you, what is the rank of the word BCA? The answer
is 4th.
10. GAP METHOD
The number of permutations of n different things taken all at
a time when m specified things, no two of which are to occur
together is n + 1 P m
¥ n!.
11. CIRCULAR PERMUTATIONS
We have discussed earlier so far, the permutations of objects
(for things) in a row. Such types of permutations are called
linear permutations of linear arrangements. If n different
things can be arranged in a row, the linear arrangements is
n!, whereas every linear arrangements have a beginning and
end but in circular permutations, there is neither beginning
nor end.
When clockwise and anti-clockwise orders are taken as
different, the number of circular permutations of n different
things taken all at a time is (n – 1)!
But, when the clockwise and anti-clockwise orders are
not different, i.e. the arrangements of beads in a necklace,
arrangements of flowers in a garland, etc., the number of circular
permutations of n different things = 1 ¥ ( n - 1)! .
2
12. RESTRICTED CIRCULAR PERMUTATIONS
(i) If clockwise and anti-clockwise arrangements are taken
as different, the number of circular permutations of
n different things, taken r at a time is given by
n
P
= r
r
(ii) If clockwise and anti-clockwise arrangements are not
taken as different, the number of circular permutations
of n different things, taken r at a time
n
P
= r
2 r
13. COMBINATIONS
The different groups or selections of a number of things taken
some or all of them at a time are called combinations.
The number of combinations of n different things taken r at
a time is denoted by n n n!
C r
and is defined as Cr
=
,
r! ¥ ( n - r)!
where r £ n, r Œ W, n Œ N.
14. SOME IMPORTANT RESULTS TO REMEMBER
(i) n C r
= nC n – r
(ii) n C x
= n C y
fi x = y or x + y = n
(iii) n C r
+ n C r – 1
= n + 1 C r
n
Cr
n- r + 1
(iv) =
n
C -
r
r
1
n n n-1 n( n-
1) n-2
(v) C = ¥ C = ¥ C
r
r( r - 1)
and so on.
(vi) n C 0
+ n C 1
+ n C 2
+ … + n C n
= 2 n
(vii)
r r-1 r-2
n 2 n 2 n 2 n 2 2n
0 + 1 + 2 + + n =
C C C C C
15. RESTRICTED COMBINATIONS
(i) The number of selections of r objects out of n different
objects when
(a) k particular objects are always included = n – k C r – k
(b) k particular objects are never included = n – k C r
(ii) The number of combinations of r things out of n different
things such that k particular things are not included
in any selection = n C r
– n – k C – k
(iii) The number of selections of r consecutive things out of
n things in a row = n – r +1
(iv) The number of selections of r consecutive things out of
Ïn : r < n
n things along a circle = Ì
Ó1:
r = n
n