P and c session 3

1. Alosies George
IIM Calcutta
alosies@gmail.com
+91- 9985-372-371
www.georgeprep.com
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Permutations and Combinations concept session 3

2. Grouping
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Grouping
Of distinct
objects
Into groups of
equal size
Division
Distribution
Into groups of
unequal size
Division
Distribution
Of identical
objects
Into groups of any size

4. Grouping of Distinct Objects
No of ways of dividing distributing
(p+q+r) distinct things
into/among 3 groups of size p,
q and r respectively.
pq things into/among p groups
of q things each
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5. Problems
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 Number of ways of dividing 12 children into 3 groups of 5, 4 and 3 children.
 Number of ways of assigning 12 students to the three sections of a class such that 5 children go to one section, 4 and
3 to the other two sections.
 Number of ways of dividing 12 children into 3 groups of 4 children each.
 Number of ways of assigning 12 students to the three sections of a class such that each section is assigned 4 students.
Concept Check
 Number of ways of assigning 12 students to the three sections of a class such that each section may be assigned with
any number of students.

6. Grouping of Identical Objects
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Number of way of distributing n identical objects among r people
(𝑛+𝑟−1)
𝑐(𝑟−1)

7. Problems - Selections not adjacent to each other
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10 soldiers are standing in a row. In how many ways two of them can be selected such that no two of them adjacent
to each other are selected?

8. Problems
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1. In how many ways can 7 identical rings be worn in 4 fingers of one hand assuming any of number of
rings can be worn in any of the fingers?
2. In how many ways can 7 different rings be worn in four particular fingers?

9. Identical vs distinct
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Boxes
Identical Distinct
Balls
Identical
Distinct
a balls and b boxes

10. Identical vs distinct
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Boxes
Identical Distinct
Balls
Identical Cases with one way each
(𝑛+𝑟−1)
𝑐(𝑟−1)
Distinct
Cases with different ways each
(Grouping) ba
a balls and b boxes

11. Points, Lines and Figures
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12. Problems – Points of Intersection
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What is the maximum number of points of intersections of
i. 10 straight lines
ii. 10 circles
iii. 10 circles and 10 straight lines

13. Problems – Figures from Points
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A plane contains 10 points. No three points are collinear.
a. How many lines can be formed passing through any two points on the plane.
b. How many triangles can be formed passing through any three points on the plane.
c. How many circles can be formed passing through any three points on the plane.
A plane contains 10 points. 5 points are collinear and no other three points are collinear.
a. How many lines can be formed passing through any two points on the plane.
b. How many triangles can be formed passing through any three points on the plane.
c. How many circles can be formed passing through any three points on the plane.

14. Problems – Quadrilateral formation
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A plane contains two sets of parallel lines with 10 lines and 12 lines respectively. If the first line in the first
set is perpendicular to the second line in the second set, then how many of the following kinds of figures
are formed.
a. Quadrilaterals
b. Rectangles
c. Squares

15. Problems – Pathway
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10 straight lines are drawn in a plane, what is the maximum number of parts into which the plane can be divided
into?
In how many ways can one move from point A to point B if each path can only be along the grid and should be the
shortest possible?
A
B