Q7.

(a). How many 4 digits number can be formed from 6 digits 1, 2, 3, 4, 5 ,6 if repetitions are not allowed. How many of these numbers are less than 4000? How many are odd? (5 Marks)

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Without Repetition, Total number can be formed

= P(6,4)

= 6!/2!

= 360 ways

For the number to be less then 4000, the leftmost digit can only be 1,2 or 3.

So the total number of numbers less the 4000 will be:

= 3*P(5,3)

= 180 ways

similarly, The total number of odd numbers

= 4 * P(5,3)

= 240 ways

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(b). What is pigeonhole principle? Explain the applications of pigeonhole principle with example? (5 Marks)

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Pigeonhole Principle: the general rule states when there are k pigeonholes and there are k+1 mails, then they will be 1 pigeonhole with at least 2 mails. A more advanced version of the principle will be the following: If mn + 1 pigeons are placed in n pigeonholes, then there will be at least one pigeonhole with m + 1 or more pigeons in it.

The Pigeonhole Principle sounds trivial but its uses are deceiving astonishing! Thus, in our project, we aim to learn and explore more about the Pigeonhole Principle and illustrate its numerous interesting applications in our daily life.

or If m pigeons occupy n pigeonholes, Where m>n, then there is at least one pigeonhole with two or more pigeons in it.

Example: Pigeonhole Principle and the Birthday problem We have always heard of people saying that in a large group of people, it is not difficult to find two persons with their birthday on the same month. For instance, 13 people are involved in a survey to determine the month of their birthday. As we all know, there are 12 months in a year, thus, even if the first 12 people have their birthday from the month of January to the month of December, the 13th person has to have his birthday in any of the month of January to December as well. Thus, we are right to say that there are at least 2 people who have their birthday falling in the same month.

In fact, we can view the problem as there are 12 pigeonholes (months of the year) with 13 pigeons (the 13 persons). Of course, by the Pigeonhole Principle, there will be at least one pigeonhole with 2 or more pigeons!

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