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PIGEONHOLE PRINCIPLE
INTRODUCTION:
Suppose there are 10 pigeons need to be arrange in 9 pigeonholes. here there are 10 pigeons but only 9 pigeonholes. Then atleast one of these pigeonholes must have atleast two pigeons in it. This is what Pigeonhole principle is. It states that if there are more pigeons than pigeonholes then there must be atleast one pigeonhole with atleast two pigeons in it. This principle is used in Probability, Geometry, Number theory etc.
Generalised Pigeon Hole Principle:-
STATEMENT: If N objects are placed into k boxes, where k ∈ Z+, then there is atleast one box containing atleast ⌈Nk ⌉ objects.
PROOF: let us suppose that each box must have strictly less than ⌈Nk ⌉ objects i.e,⌈Nk ⌉ − 1 objects. Now there are k boxes and each box has objects ≤ ⌈Nk ⌉ − 1 =⇒ total no.of objects ≤ k
⌈Nk ⌉ − 1
=⇒ N ≤ k
⌈Nk ⌉ − 1
=⇒ N ≤ k
⌈Nk ⌉ − 1
< k.Nk [∵ ⌈x⌉ − 1 < x] = N =⇒ N < N which is contradiction. ∴ atleast one of box containing atleast ⌈Nk ⌉ objects.
PROBLEMS:
Q1: What is the minimum no. of people need to ensure that two people share the same birthday (in leap year)?
A : No.of days in a leap year (k)= 366 let minimum no.of people need be N. Given that two people share the same birthday i.e, ⌈ 366 N ⌉ = 2 =⇒ N 366 −^1 = (2 − 1) =⇒ N − 1 = 366 =⇒ N = 366 + 1 = 367 ∴ The minimum no. of people needed to ensure that two people share the same birthday is 367
Q2: What is the minimum no. of words need(using English Language) to ensure that two words start with the same letter?
A : No.of letters in English language (k)= 26 let minimum no.of words needed be N. Given that two words start with same letter i.e, ⌈ 26 N ⌉ = 2 =⇒ N 26 − 1 = (2 − 1) =⇒ N − 1 = 26 =⇒ N = 26 + 1 = 27 ∴ The minimum no. of words needed to ensure that two words start with the same letter is 27
Q3: Among 50 people, what is the minimum no.of people that must be born on same month?
A : No.of months per year (k)= 12 No.of people (N) = 50 then, ⌈Nk ⌉ = ⌈^5012 ⌉ =⇒ ⌈Nk ⌉ = ⌈ 4. 16 ⌉ =⇒ ⌈Nk ⌉ = 5 ∴ The minimum no. of people must be born on same month is 5
Q4: How many students in a class must there be to ensure that 3 students get the same grade (one of A,B,C,D or E)?
A : No.of grades (k)= 5 let minimum no.of words needed be N. Given that three got the same grade i.e, ⌈N 5 ⌉ = 3 =⇒ N^ − 5 1 = (3 − 1) =⇒ N − 1 = 5 X 2 =⇒ N = 10 + 1 = 11 ∴ The minimum no. of students must be there ensure that three students get the same grade is 11
Q5: A bowl contains 10 red and 10 yellow balls. How many balls must be selected to ensure that 3 balls get the same color?
A : No.of types of colors of balls (k)= 2 let no.of balls selected be N. Given that three balls of same color i.e, ⌈N 2 ⌉ = 3 =⇒ N^ − 2 1 = (3 − 1) =⇒ N − 1 = 2 X 2 =⇒ N = 4 + 1 = 5 ∴ The no. of balls must be selected ensure that three balls of same color is 5
Q6: What is the minimum no.of students, each of whom comes from one of the 28 states or 8 union territories must be enrolled in a university to guarantee that there are atleast 50 who come from the state?
A : No.of states and UTs (k)= 36(28+8) let no.of students enrolled be N. Given that 50 students comes from same state i.e, ⌈ N 36 ⌉ = 50 =⇒ N 36 − 1 = (50 − 1) =⇒ N − 1 = 36 X 49 =⇒ N = 1764 + 1 = 1765 ∴ The no. of students must be enrolled to ensure that there are atleast 50 students are of from same state is 1765
now make the square into 4 equal regions as below
it is observed that the distance between p and q is
√ 2
then the distance between any two points in a region is less than
√ 2
Here 5 points are drawn from inside of square then those point must be drawn from these 4 regions. No.of regions (k)= 4 No.of points drawn (N) = 5 then, ⌈Nk ⌉ = ⌈^54 ⌉
=⇒ ⌈Nk ⌉ = ⌈ 1. 25 ⌉
=⇒ ⌈Nk ⌉ = 2
∴ there are atleast 2 points with distance less than
√ 2
