document.write( "Question 1196153: In an​ orchestra, 22 people can play stringed​ instruments, 21 can play​ brass,14 and can play percussion.​ Further, 8 of the performers can play both strings and​ brass, whereas 4 can play both strings and percussion. If no one can play all three types of​ instruments, what are the maximum and minimum numbers of people in the​ orchestra? \n" ); document.write( "
Algebra.Com's Answer #828868 by ikleyn(52867)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "In an​ orchestra, 22 people can play stringed​ instruments, 21 can play​ brass,
\n" ); document.write( "and 14 can play percussion.​
\n" ); document.write( "Further, 8 of the performers can play both strings and​ brass,
\n" ); document.write( "whereas 4 can play both strings and percussion.
\n" ); document.write( "If no one can play all three types of​ instruments,
\n" ); document.write( "what are the maximum and minimum numbers of people in the​ orchestra?
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\n" ); document.write( "\n" ); document.write( "Tutor @MathTherapy in his post provided more advanced analysis,
\n" ); document.write( "which gave better estimations than my previous solution. \r
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document.write( "    (notice that his estimations do not disprove mine: they simply are \"highlight%28more_advanced%29\" estimations).\r\n" );
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\n" ); document.write( "\n" ); document.write( "After reading his post, I got understanding where my solution can be improved,
\n" ); document.write( "so I place here new version, which gives the answer close to that of the @MathTherapy post.\r
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\n" ); document.write( "\n" ); document.write( "Thus, below you see my improved solution, and after it, I still keep my previous, now obsolete version,
\n" ); document.write( "so a reader can compare and make conclusions for himself (or herself) - it might be educational.\r
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\n" ); document.write( "\n" ); document.write( "I could delete my solution, at all - but still think that it can be useful for somebody,
\n" ); document.write( "who wants to see the solution to this problem based on Inclusive-Exclusive principle.\r
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\n" ); document.write( "\n" ); document.write( "             Thanks to tutor @MathTherapy for really nice job !\r
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        - - - Updated version - - -

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document.write( "We have a universal set U of all people of the orchestra, and three its subsets\r\n" );
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document.write( "    S (stringed instruments) of 22 persons\r\n" );
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document.write( "    B (brass)                of 21 persons\r\n" );
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document.write( "    P (percussion)           of 14 persons.\r\n" );
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document.write( "We have information about intersections\r\n" );
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document.write( "    n(S ∩ B) = 8;\r\n" );
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document.write( "    n(S ∩ P) = 4,\r\n" );
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document.write( "    n(S ∩ B ∩ P) = 0.\r\n" );
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document.write( "Write the inclusion-exclusion principle formula\r\n" );
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document.write( "    n(S U B U P) = n(S) + n(B) + n(P) - n(S ∩ B) - n(S ∩ P) - n(B ∩ P) + n(S ∩ B ∩ P) =\r\n" );
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document.write( "                 =  22  +  21  +  14  -    8     -    4     - n(B ∩ P) + 0,\r\n" );
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document.write( "or\r\n" );
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document.write( "    n(orchestra) = 45 - n(B ∩ P).\r\n" );
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document.write( "In this equality, the value of the term n(B ∩ P) is not given and is not known.\r\n" );
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document.write( "We only know that n(B ∩ P) is not greater than  n(B) - n(B ∩ S) = 21-8 = 13       <<<---===  my corrections start here\r\n" );
document.write( "and not greater than n(P) - n(S ∩ P) = 14-4 = 10.\r\n" );
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document.write( "So, n(B ∩ P) is between 0 and 10 (inclusive).\r\n" );
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document.write( "Therefore, n(orchestra) is not greater than 45 and not lesser than 45-10 = 35.\r\n" );
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document.write( "ANSWER.  The number of people in the orchestra is not lesser than 35 and not greater than 45.\r\n" );
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\n" ); document.write( "\n" ); document.write( "                At the end,  I have one more notice to a reader and to tutor @MathTherapy.\r
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\n" ); document.write( "\n" ); document.write( "In the analysis, we absolutely correctly arrive to the conclusion that  n(B ∩ P)  is between  0  and  10.\r
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\n" ); document.write( "\n" ); document.write( "But we have no reasons to reject the end-point possibilities  n(B ∩ P) = 0  or  n(B ∩ P) = 10.\r
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\n" ); document.write( "\n" ); document.write( "Therefore, my answer   35 <= n(orchestra) <= 45   is still different \r
\n" ); document.write( "\n" ); document.write( "from the @MathTherapy's answer   36 <= n(orchestra) <= 44.\r
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        - - - My previous, now obsolete version - - -

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\r\n" );
document.write( "We have a universal set U of all people of the orchestra, and three its subsets\r\n" );
document.write( "\r\n" );
document.write( "    S (stringed instruments) of 22 persons\r\n" );
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document.write( "    B (brass)                of 21 persons\r\n" );
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document.write( "    P (percussion)           of 14 persons.\r\n" );
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document.write( "\r\n" );
document.write( "We have information about intersections\r\n" );
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document.write( "    n(S ∩ B) = 8;\r\n" );
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document.write( "    n(S ∩ P) = 4,\r\n" );
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document.write( "    n(S ∩ B ∩ P) = 0.\r\n" );
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document.write( "Write the inclusion-exclusion principle formula\r\n" );
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document.write( "    n(S U B U P) = n(S) + n(B) + n(P) - n(S ∩ B) - n(S ∩ P) - n(B ∩ P) + n(S ∩ B ∩ P) =\r\n" );
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document.write( "                 =  22  +  21  +  14  -    8     -    4     - n(B ∩ P) + 0,\r\n" );
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document.write( "or\r\n" );
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document.write( "    n(orchestra) = 45 - n(B ∩ P).\r\n" );
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document.write( "\r\n" );
document.write( "In this equality, the value of the term n(B ∩ P) is not given and is not known.\r\n" );
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document.write( "We only know that n(B ∩ P) is not greater than the minimum min(21,14) = 14, \r\n" );
document.write( "so we know that n(B ∩ P) is between 0 and 14.\r\n" );
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document.write( "Therefore, n(orchestra) is not greater than 45 and not lesser than 45-14 = 31.\r\n" );
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document.write( "ANSWER.  The number of people in the orchestra is not lesser than 31 and not greater than 45.\r\n" );
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