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Chapter 6: Q11SE (page 440)

A fortune cookie company makes \({\bf{213}}\) different fortunes. A student eats at a restaurant that uses fortunes from this company and gives each customer one fortune cookie at the end of a meal. What is the largest possible number of times that the student can eat at the restaurant without getting the same fortune four times?

Short Answer

Expert verified

As a result, we can only visit \(639\) times without receiving the same luck four times.

Step by step solution

01

Definition of pigeonhole principle

The pigeonhole principle asserts that if n things are placed in m containers, with n>m, at least one container must contain multiple items.

02

Solution

Let us simplify,

\(k = 213\)

Since we want at most\(3\)objects per box (fortune cookie), the generalized pigeonhole principle then gives us:

\((N/k) = 3\)

since\(k = 213\):

\((N/213) = 3\)

The largest possible value for\(N\)such that the above equality is true is then the value for which\(N/213\)is an integer (by the definition of the ceiling function)

Multiply each side of the equation by\(3\):

\(N = 3 \cdot 213 = 639\)

Therefore, we can visit at most \(639\) times without getting the same fortune four times.

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Most popular questions from this chapter

In this exercise we will count the number of paths in the\(xy\)plane between the origin \((0,0)\) and point\((m,n)\), where\(m\)and\(n\)are nonnegative integers, such that each path is made up of a series of steps, where each step is a move one unit to the right or a move one unit upward. (No moves to the left or downward are allowed.) Two such paths from\((0,0)\)to\((5,3)\)are illustrated here.

a) Show that each path of the type described can be represented by a bit string consisting of\(m\,\,0\)s and\(n\,\,1\)s, where a\(0\)represents a move one unit to the right and a\(1\)represents a move one unit upward.

b) Conclude from part (a) that there are \(\left( {\begin{array}{*{20}{c}}{m + n}\\n\end{array}} \right)\) paths of the desired type.

How can you find the number of bit strings of length ten that either begin with 101 or end with 010 ?

a) Derive a formula for the number of permutations ofobjects of k different types, where there aren1 indistinguishable objects of type one,n2 indistinguishable objects of type two,..., andnk indistinguishable objects of type k.

b) How many ways are there to order the letters of the word INDISCREETNESS?

When the numbers from \({\rm{1}}\) to \({\rm{1000}}\) are written out in decimal notation, how many of each of these digits are used?

a) \({\rm{0}}\)

b) \({\rm{1}}\)

c) \({\rm{2}}\)

d) \({\rm{9}}\)

Let\(n\)and \(k\) be integers with \(1 \le k \le n\). Show that

\(\sum\limits_{k = 1}^n {\left( {\begin{array}{*{20}{l}}n\\k\end{array}} \right)} \left( {\begin{array}{*{20}{c}}n\\{k - 1}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}{2n + 2}\\{n + 1}\end{array}} \right)/2 - \left( {\begin{array}{*{20}{c}}{2n}\\n\end{array}} \right)\)
See all solutions

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