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Counting and Set Theory

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How many ways can n books be placed on k distinguishable shelves?
a) if the books are indistinguishable copies of the same title?
b) if no two books are the same, and the positions of the books on the shelves matter?

How many ways are there to deal bands of five cards to each of six players from a deck containing 48 different cards?

How many ways are there to select three unordered elements from a set with five elements when repetition is allowed?

There are 2504 computer science students at a school. Of these, 1876 have taken a course in Java, 999 have taken a course in Linux, and 345 have taken a course in C. Further, 876 have taken courses in both Java and Linux, 231 have taken courses in both Linux and C, and 290 have taken courses in both Java and C. If 189 of these students have taken courses in Linux, Java and C, how many of these 2504 students have not taken a course in any of these three programming languages?

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a. If the books are indistinguishable copies of the same title?
Solution:
There are k distinguishable shelves and n books are to be placed in them.
If the shelves are represented by space between two sticks | | and books are represented by *, then
| | * * * | | * * | | | ..... | (Total of k + 1 sticks for k shelves and n *s for n books)
Assume that the first and last sticks are fixed. Hence there are (k+1-2) + n items between the first and last sticks. Out of these n books have to be placed. Hence the number of ways = (k + 1 - 2 + n) C (n)
= (n + k - 1) C (n)

b. If no two books are the same, and the positions on the shelves matter?
Solution:
There are k distinguishable shelves and n distinguishable books.
For each book, there are k possible ...

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A Set Theory in Real Analysis

Formal Math Proofs

Prove that each of the following sets is countable:

a) The set of all numbers with two distinct decimal expansions (like 0.500... and 0.4999...);

b) The set of all rational points in the plane (i.e., points with rational coordinates);

c) The set of all rational intervals (i.e., intervals with rational end points);

d) The set of all polynomials with rational coefficients.

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