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Chapter 5: Problem 2

Show that if eight people are in a room, at least two of them have birthdays that occur on the same day of the week.

Short Answer

Expert verified
By applying the Pigeonhole Principle, it can be proven that if eight people are in a room, at least two of them have birthdays occurring on the same day of the week.

Step by step solution

01

Understanding the Problem

The problem is to prove that if there are eight people in a room, at least two of them have birthdays on the same day of the week. The crucial aspect to understand here is that there are only seven days in a week.
02

Application of Pigeonhole Principle

According to the Pigeonhole Principle, if there are eight 'items' (eight people with birthdays in this case) and they are put into seven 'containers' (the seven days of the week), then at least one container must contain more than one item. Therefore, at least one day of the week must be the birthday for more than one person.

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Most popular questions from this chapter

If \(n \in \mathbf{Z}^{+}\)with \(n \geq 4\), verify that \(S(n, n-2)=\left(\begin{array}{l}n \\ 3\end{array}\right)+\) \(3\left(\begin{array}{l}n \\ 4\end{array}\right)\).

For the access function developed in Example \(5.10(\mathrm{~d})\), the matrix \(A=\left(a_{i j}\right)_{m \times n}\) was stored in a one-dimensional array using the row, major implementation. It is also possible to store this matrix using the column major implementation, where each entry \(a_{i i}, 1 \leq i \leq m\), in the first column of \(A\) is stored in locations \(1,2,3, \ldots, m\), respectively, of the array, when \(a_{11}\) is stored in location 1 . Then the entries \(a_{a 2}, 1 \leq i \leq m\), of the second column of \(A\) are stored in locations \(m+1, m+2, m+3, \ldots, 2 m\), respectively, of the array, and so on. Find a formula for the access function \(g\left(a_{i}\right)\) under these conditions.

Use the fact that every polynomial equation having real-number coefficients and odd degree has a real root in order to show that the function \(f: \mathbf{R} \rightarrow \mathbf{R}\), defined by \(f(x)=\) \(x^{5}-2 x^{2}+x\), is an onto function. Is \(f\) one-to-one?

For \(n \in \mathbf{Z}^{+}\), define \(\tau: \mathbf{Z}^{+} \rightarrow \mathbf{Z}^{+}\)by \(\tau(n)=\) the number of positive-integer divisors of \(n\). a) Let \(n=p_{1}^{\rho_{1}} p_{2}^{2} p_{3}^{e_{3}} \cdots p_{k}^{6 k}\), where \(p_{1}, p_{2}, p_{3}, \ldots, p_{k}\) are distinct primes and \(e_{i}\) is a positive integer for all \(1 \leq i \leq k\). What is \(\tau(n)\) ? b) Determine the three smallest values of \(n \in \mathbf{Z}^{*}\) for which \(\tau(n)=k\), where \(k=2,3,4,5,6\). c) For any \(k \in \mathbf{Z}^{+}, k>1\), prove that \(\tau^{-1}(k)\) is infinite. d) If \(a, b \in \mathbf{Z}^{+}\)with \(\operatorname{gcd}(a, b)=1\), prove that \(\tau(a b)\) \(=\tau(a) \tau(b)\).

Let \(A \subset\\{1,2,3, \ldots, 25\\}\) where \(|A|=9\). For any subset \(B\) of \(A\) let \(s_{B}\) denote the sum of the elements in \(B\). Prove that there are distinct subsets \(C, D\) of \(A\) such that \(|C|=|D|=5\) and \(s_{C}=s_{D}\).
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