document.write( "Question 1143356: A school has 731 students. Prove that there must be at least three students that have the same birthday assuming that no one has a birthday on Feb 29.
\n" ); document.write( "Can anybody please help us with this question?
\n" ); document.write( "We would be very grateful
\n" ); document.write( "Thank you💜 \n" ); document.write( "

Algebra.Com's Answer #764238 by ikleyn(52786) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "There is so called  \"the pigeonhole principle\"  in Math:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    If 7 pigeons are placed in 6 holes, then at least one hole contains 2 or more pigeons. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In this joking form it is obvious and does not require more detailed proofs / explanations.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Further, in more general form, \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    if (n+1) pigeons are placed in \"n\" holes, then there is at least one hole containing 2 or more pigeons. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In this form it is obvious, again, and does not require more detailed proofs / explanations.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now let me formulate even more general THEOREM.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    If there are  n*m+1  items in  \"m\" containers, then there is at least one container containing  \"n+1\"  or more  items.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The proof is in three lines.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    If there is NO such a container, then the total number of items in \"m\" containers is NOT MORE than  n*m.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    It  CONTRADICTS  to the given part (!)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    This contradiction proves the statement.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now let's return to our problem.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We have 731 = 2*365 + 1  students  (= items),  and\r\n" );
document.write( "\r\n" );
document.write( "        365 days in the year (that are \"containers\" in this case).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From the Theorem, there is at least one container containing (2+1) = 3 or more items.\r\n" );
document.write( "\r\n" );
document.write( "In other words, there is at least one day (one date in an year), when 3 or more students celebrate their birthdays.\r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-----------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In MATHEMATICS (notice all letters are capital, which means that I am talking about TRUE Math)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "this principle is called \"the Dirichlet's principle\", after the famous mathematician Johann Peter Gustav Lejeune Dirichlet
\n" ); document.write( "(1805 - 1859).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "About Dirichlet, see these Internet articles\r
\n" ); document.write( "\n" ); document.write( "http://www-history.mcs.st-and.ac.uk/Biographies/Dirichlet.html\r
\n" ); document.write( "\n" ); document.write( "https://en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlet\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "About \"the Dirichlet's principle\", see this Internet article\r
\n" ); document.write( "\n" ); document.write( "https://en.wikipedia.org/wiki/Pigeonhole_principle\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );