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Bijections

Bijections, surjections, and injections are three types of functions which associate the elements between two sets. For example, each word in this sentence can be mapped to exactly one in the last.

Problem Solving - Basic

         

Suppose that \(10\) dice are rolled. Each die is a regular \(6\)-sided die with numbers \(1\) through \(6\) labelled on the sides. How many different distinct sums of all 10 numbers are possible?

Details and assumptions

The sums \(5 = 1+4 \) and \(5 = 2 + 3 \) are the same sum (namely 5), and should only be counted once.

Let \(A=\{1, 2, \ldots, 6\}\) and \(B=\{b_1, b_2, \ldots , {b}_{5}\},\) where the \(b_i\) are all distinct. How many distinct functions from \(A\) to \(B\) are possible?

In New York city, all the streets are arranged in a grid. A hospital is located 5 blocks east and 6 blocks north of an accident. How many ways are there to get there, if we only go 1 block north or 1 block east at each intersection?

Suppose that \(10\) dice are rolled. Each die is a regular \(6\)-sided die with numbers \(1\) through \(6\) labelled on the sides. How many different distinct sums of all 10 numbers are possible?

Details and assumptions

The sums \(5 = 1+4 \) and \(5 = 2 + 3 \) are the same sum (namely 5), and should only be counted once.

Mary has enrolled in \(6\) courses: Chemistry, Physics, Math, English, French and Biology. She has one textbook for each course and wants to place them on a shelf. How many ways can she arrange the textbooks so that the English textbook is placed before the French textbook?

Details and assumptions

If the books are ordered from left to right, then the "English textbook is placed before the French textbook" means that the English textbook was placed to the left of the French textbook. They do not need to be immediately beside each other.

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