1.Algebra Booster

Permutations and Combinations 5.5
= Co-efficient of x n – m in (1 – mx n + … ) ¥ (1 – x) –m
= m + n – m – 1 C n – m
= n – 1 C m – 1
(ii) The number of negative integral solutions of the equation
x 1
+ x 2
+ … + x m
= n.
Hence, the required number of solutions
= Co-efficient of x n in (1 + x + x 2 + … + x n ) m
Ê
n
1 - x
= Co-efficient of x n in Á
Ë 1 - x
+ 1
m
= Co-efficient of x n in (1 – x n + 1 ) m ¥ (1 – x) –m
= Co-efficient of x n (1 – mx n + 1 + … ) ¥ (1 – x) –m
= m + n – 1 C m – 1
(iii) The number of integral solutions of the equation
x 1
+ x 2
+ … + x m
= n, when x 1
≥ c 1
, x 2
≥ c 2
, …, x m
≥ c m
Let y 1
= x 1
– c 1
, y 2
= x 2
– c 2
, …, y n
= x n
– c n
Then the given equation can be written as
y 1
+ y 2
+ y 3
+ … + y m
= n – (c 1
+ c 2
+ c 3
+ … + c m
)
where y i
≥ 0, i = 1, 2, 3, …, n
Hence, the required number of solutions
= the number of solutions of the negative integers
Ê
m
ˆ
n-1
= Án- Âci˜
+ Cm-1
Ë i = 1 ¯
26. SELECTION OF SQUARES
Let there be m rows, whereas 1st row has m 1
squares, 2nd row
has m 2
squares, 3rd row has m 3
squares and so on.
Now, if we placed n x’s in the squares such that each row
contain at least one x, the number of the possible ways
= Co-efficient of x n in
m1 m1 2 m
(
1 m
Cx+ C x + + C x
1)
ˆ
˜
¯
1 2
m1
m2 m2 2 m2 m
Cx
2
1 C2x Cm
2
m3 m3 2 m3 m
Cx
3
1 C2x Cm
x
3
¥ ( + + + x )
¥ ( + + + )
¥
27. GEOMETRICAL PROBLEMS
(i) If there are n points in a plane of which m (< n) are
colinear, the total number of straight lines obtained by
joining these n points is n C 2
– m C 2
+ 1.
(ii) If there are n points in a plane of which m (< n) are
colinear, the total number of triangles formed by joining
these n points is n C 3
– m C 3
.
(iii) The number of diagonals in a polygon of n sides
n nn ( - 3)
= C2
- 2 = .
2
(iv) If n straight lines are drawn in a plane such that no
two lines are parallel and no three lines are concurrent,
the number of parts into which these lines divided the
plane is 1 + Sn.
(v) If m parallel lines in a plane are intersected by a family
of lines other n parallel lines, the total number of parallelograms
so formed
= m C 2
¥ n C 2
mn( m -1)( n -1)
=
4
(vi) Number of squares: The number of squares of any
size n ¥ p that can be formed
= np + (n – 1)(p – 1) + (n – 2)(p – 2) + …
+ [n(n – 1)(p – (n – 1))]
Ï
Ô
Ô
n
( n- r + 1)( p - r + 1): n<
p
r = 1
= Ì
n
Ô 2
 :
=
r = 1
Ô
Ó
Â
r n p
(vii) Number of rectangles: The number of rectangles of
any size n ¥ p that can be formed
= n + 1 C 2
¥ p + 1 C 2
Ïnp( n + 1)( p + 1) : n<
p
Ô 4
= Ì
2
Ô Ênn
( + 1) ˆ
Á ˜ : n=
p
ÔÓ Ë 2 ¯
EXERCISES
Level I
FACTORIAL NOTATION
x x 1
1. Find x, if + = .
5! 6! 7!
(2 n)!
2. Simplify: .
n!
(Questions based on Fundamentals)
3. Simplify: 3 ◊ 6 ◊ 9 ◊ 12…(3 n - 3)3 n
.
3 n
4. Find the unit digit of 1! + 2! + 3! + 4! + … + (10)!.
5. If N = a! + b! + c! + d! + e! be a two-digit number, find
the value of N, where a, b, c, d, e ΠI.
6. If the product of factorials of n consecutive positive
integers be a single-digit number, find the maximum
value of n.