document.write( "Question 1158217: A printer is printing out 3-digit numbers between 100-999 such that the digits are not repeated.
\n" ); document.write( "(a) If 2000 such numbers are printed out, at least how many of them will be identical? Justify your
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\n" ); document.write( "(b) At least how many of the numbers should be printed, so that at least 3 of them will be identical?
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\n" ); document.write( "(c) At least how many of the numbers should be printed, so that the number 243 is printed at least 6
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Algebra.Com's Answer #781145 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "A printer is printing out 3-digit numbers between 100-999 such that the digits are not repeated.
\n" ); document.write( "(a) If 2000 such numbers are printed out, at least how many of them will be identical? Justify your answer.
\n" ); document.write( "(b) At least how many of the numbers should be printed, so that at least 3 of them will be identical? Justify your answer.
\n" ); document.write( "(c) At least how many of the numbers should be printed, so that the number 243 is printed at least 6 times? Justify your answer.
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document.write( "First, we need estimate, how many such 3-digit numbers between 100 and 999 (inclusive) do exist\r\n" );
document.write( "such that the digits are not repeated.\r\n" );
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document.write( "This maximal possible amount of such numbers is 9*8*7 = 504.\r\n" );
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document.write( "    (any of 9 digits from 1 to 9 in the 1-st position;\r\n" );
document.write( "     any of 8 remaining digits in the   2-nd position; and\r\n" );
document.write( "     any of 7 remaining digits in the   3-rd position).\r\n" );
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document.write( "(a)  If 2000 such numbers are printed out, at least how many of them will be identical ?/.\r\n" );
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document.write( "     Now, I state that of 2000 such printed numbers, there are at least 3 (three) identical numbers.\r\n" );
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document.write( "     Indeed, if any printed number is repeating less than 3 times, than the total amount of printed instances     \r\n" );
document.write( "     would be not more than 3*504 = 1512;  but in reality, we have 2000 (!) printed numbers.\r\n" );
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document.write( "     Thus the ANSWER to question (a) is 3.\r\n" );
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document.write( "(b)  At least how many of the numbers should be printed, so that at least 3 of them will be identical ?\r\n" );
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document.write( "     From the logic of my solutions to (a), it is clear, that at least 3*504+1 = 1513 such numbers should be printed, \r\n" );
document.write( "     so that at least 3 of them will be identical. \r\n" );
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document.write( "     Thus the ANSWER to question (b) is 3*504+1 = 1513.\r\n" );
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document.write( "(c)  At least how many of the numbers should be printed, so that the number 243 is printed at least 6 times ?\r\n" );
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document.write( "     This question is posed INCORRECTLY:  infinitely many such numbers can be printed, NO ONE of which is the number 243.\r\n" );
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\n" ); document.write( "\n" ); document.write( "The principle of proving such statements/(solving such problems) is called \"the pigeonhole principle\".\r
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\n" ); document.write( "\n" ); document.write( "See the lesson\r
\n" ); document.write( "\n" ); document.write( " - The \"pigeonhole principle\" problems \r
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