Question

There are \({\rm{12}}\) signs of the zodiac. How many people are needed to guarantee that at least six of these people have the same sign?

Short answer

The number of people that are needed to guarantee that at least six of the people have the same sign is \({\rm{61}}\).

Step-by-step solution

Concept Introduction

Generalized pigeonhole principle When\(N\)objects are place into\(k\)boxes, then there exists a box with at least\(\left( {{{\rm{N}} \mathord{\left/

{\vphantom {{\rm{N}} {\rm{k}}}} \right.

\kern-\nulldelimiterspace} {\rm{k}}}} \right)\)objects.

Calculation for number of People

The value for number of boxes is\[{\rm{k = 12}}\).

Since at least 6 objects per box is needed, the generalized pigeonhole principle then gives that –

\[\left[ {{{\rm{N}} \mathord{\left/

{\vphantom {{\rm{N}} {\rm{k}}}} \right.

\kern-\nulldelimiterspace} {\rm{k}}}} \right){\rm{ = 6}}\)

Substituting the value –

\[\left[ {{{\rm{N}} \mathord{\left/

{\vphantom {{\rm{N}} {{\rm{12}}}}} \right.

\kern-\nulldelimiterspace} {{\rm{12}}}}} \right){\rm{ = 6}}\)

The smallest possible value for\[{\rm{N}}\)is then\[{\rm{6 - 1 = 5}}\)times the multiple of the denominator increased by\[{\rm{1}}\).

\[\begin{array}{c}{\rm{N = 5}} \cdot {\rm{12 + 1}}\\{\rm{ = 60 + 1}}\\{\rm{ = 61}}\end{array}\)

Note that:\[\left[ {{{60} \mathord{\left/

{\vphantom {{60} {{\rm{12}}}}} \right.

\kern-\nulldelimiterspace} {{\rm{12}}}}} \right){\rm{ = 5}}\)and\[\left[ {{{{\rm{61}}} \mathord{\left/

{\vphantom {{{\rm{61}}} {{\rm{12}}}}} \right.

\kern-\nulldelimiterspace} {{\rm{12}}}}} \right)\left[ {{\rm{5 + 1/12}}} \right){\rm{ = 6}}\). Thus, at least\[{\rm{61}}\)people are needed to guarantee that there are at least\[{\rm{6}}\)with the same sign.

Therefore, the result is obtained as \[{\rm{61}}\) people.