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31. How many groups of 6 persons can be formed from 8 men and 7 women?
  A.  5125
  B.  4840
  C.  5005
  D.  5250
     
   
View Answer

Problems based on number of combinations.
(i) In simple cases.
(ii) When certain things are included or excluded.
Here, Total Men = 8
Total Women = 7
So, Total number of person = 8 + 7 = 15
No. of groups = 15C6 ways
∴ Required number of groups = 15C6
=
15! / 15!(15 − 6)!

=
15! / 6! x 9!

=
15 x 14 x 13 x 12 x 11 x 10 x 9! / 6 x 5 x 4 x 3 x 2 x 1 x 9!

= 7 x 13 x 11 x 5 = 5005

32. There are 10 oranges in a basket. Find the number of ways in which 3 oranges are chosen from the basket.
  A.  155
  B.  140
  C.  100
  D.  120
     
   
View Answer

Problems based on number of combinations.
(i) In simple cases.
(ii) When certain things are included or excluded.
Here, Total oranges = 10
No. of oranges to be chosen = 3
So, Total number of ways = 10C3
=
10! / 3!(10 − 3)!

=
10! / 3! x 7!

=
10 x 9 x 8 x 7! / 3 x 2 x 1 x 7!
= 10 x 3 x 4 = 120

33. There are 25 students in a class. Find the number of ways in which a committee of 3 students is to be formed.
  A.  2300
  B.  2400
  C.  2200
  D.  2320
     
   
View Answer

Problems based on number of combinations.
(i) In simple cases.
(ii) When certain things are included or excluded.
Here, Total students = 25
A committee consists of 3 students.
So, Total number of ways = 25C3
=
25! / 3!(25 − 3)!

=
25! / 3! x 22!

=
25 x 24 x 23 x 22! / 3 x 2 x 1 x 22!
= 25 x 4 x 23 = 2300

34. Find the number of triangles fromed by joining the vertices of a polygon of 12 sides.
  A.  210
  B.  220
  C.  235
  D.  205
     
   
View Answer

Shortcut:
The number of triangles which can be formed by joining the angular points of a polygon of 'n' sides as vertices are
n(n − 1)(n − 2) / 6

Here, n = 12
Using the value in the shortcut, we get:
=
12(12 − 1)(12 − 2) / 6

=
12 x 11 x 10 / 6
= 2 x 11 x 10 = 220

35. Find the number of diagonals of a polygon of 12 sides.
  A.  58
  B.  60
  C.  50
  D.  54
     
   
View Answer

Shortcut:
The number of diagonals which can be formed by joining the vertices of a polygon of 'n' sides are
n(n − 3) / 2

Here, n = 12
Using the value in the shortcut, we get:
=
12(12 − 3) / 2

=
12 x 9 / 2
= 6 x 9 = 54

36. How many diagonals are there in a decagon?
  A.  35
  B.  38
  C.  45
  D.  42
     
   
View Answer

Shortcut:
The number of disgonals which can be formed by joining the vertices of a polygon of 'n' sides are
n(n − 3) / 2

Here, n = 10
Using the value in the shortcut, we get:
=
10(10 − 3) / 2

=
10 x 7 / 2
= 5 x 7 = 35

37. In a chess board there are 9 vertical and 9 horizontal lines. Find the number of rectangles formed in the chess board.
  A.  1230
  B.  1290
  C.  1296
  D.  1350
     
   
View Answer

Shortcut:
If there are 'a' horizontal and 'b' vertical lines, then the number of different rectangles formed are given by (aC2 x bC2).
Here, a = 9, b = 9
Using the value in the shortcut, we get:
= 9C2 x 9C2
=
9! / 2!(9 − 2)!
x
9! / 2!(9 − 2)!

=
9! / 2! x 7!
x
9! / 2! x 7!

=
9 x 8 x 7! / 2 x 1 x 7!
x
9 x 8 x 7! / 2 x 1 x 7!
= 9 x 4 x 9 x 4 = 1296

38. 10 parallel lines are intersected by 13 other parallel lines. Find the number of parallelograms thus formed.
  A.  3600
  B.  3510
  C.  3560
  D.  3480
     
   
View Answer

Shortcut:
If there are 'a' horizontal and 'b' vertical lines, then the number of different rectangles formed are given by (aC2 x bC2).
Here, a = 10, b = 13
Using the value in the shortcut, we get:
= 10C2 x 13C2
=
10! / 2!(10 − 2)!
x
13! / 2!(13 − 2)!

=
10! / 2! x 8!
x
13! / 2! x 11!

=
10 x 9 x 8! / 2 x 1 x 8!
x
13 x 12 x 11! / 2 x 1 x 11!
= 5 x 9 x 13 x 6 = 3510

39. In a party every person shakes hands with every other persons. If there was a total of 210 handshakes in the party, find the number of persons who were present in the party.
  A.  18
  B.  24
  C.  15
  D.  21
     
   
View Answer

Shortcut:
In a function every man shakes hand with every other man. If there was a total of H handshakes in the function, the number of men 'n' who were preent in the function can be calculated by the following equation.
n(n − 1) / 2
= H.
Here, H = 210
Using the value in the shortcut, we get:
n(n − 1) / 2
= 210
n(n − 1) = 2 x 210
n(n − 1) = 21 x 20
n = 21

40. Eight men entered a lounge simultaneously. If each person shook hands with the other then find the total number of handshakes.
  A.  21
  B.  25
  C.  28
  D.  30
     
   
View Answer

Shortcut:
In a function every man shakes hand with every other man. If there was a total of H handshakes in the function, the number of men 'n' who were preent in the function can be calculated by the following equation.
n(n − 1) / 2
= H.
Here, n = 8, H = ?
Using the value in the shortcut, we get:
8(8 − 1) / 2
= H
H =
8 x 7 / 2
= 4 x 7 = 28

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