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 #828979 by MathTherapy(10556) $\"\"$ $\"About$

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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?
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document.write( "which leads to:\r\n" );
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document.write( "Although not stated, the following is based on the premise that each member can play at least 1 instrument. \r\n" );
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document.write( "Let S, B, and P be the number of persons who can play stringed, brass, and percussion instruments, respectively\r\n" );
document.write( "Also, let B&P be the number that can play both brass and percussion instruments\r\n" );
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document.write( "Then: Number of persons who can play stringed instrments, ONLY: 22 - 8 - 4 = 10\r\n" );
document.write( "      Number of persons who can play brass instruments, ONLY: 21 - 8 - B&P = 13 - B&P\r\n" );
document.write( "      Number of persons who can play percussion instrments, ONLY: 14 - 4 - B&P = 10 - B&P\r\n" );
document.write( "      Total number in orchestra would then be: 10 + 8 + 4 + (13 - B&P) + B&P + (10 - B&P) = 45 - B&P\r\n" );
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document.write( "Now, 13 - B&P indicates that B&P CANNOT be 13 or more, so B&P can be 12 or less. However, 10 - B&P indicates that B&P CANNOT be 10 or\r\n" );
document.write( "more, so B&P can be 9 or less. Therefore, B&P MUST be 9 or less, which means that the MINIMUM and MAXIMUM that B&P can be are 1 and 9. \r\n" );
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document.write( "If B&P is 1, then number in orchestra = 45 - 1, or 10 + 8 + 4 + 13 - 1 + 1 + 10 - 1 = 44\r\n" );
document.write( "If B&P is 9, then number in orchestra = 45 - 9, or 10 + 8 + 4 + 13 - 9 + 9 + 10 - 9 = 36\r\n" );
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document.write( "So, the MAXIMUM number that can be in the orchestra is 44 and the MINIMUM number is 36.\r\n" );
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document.write( "Refer to the above Venn Diagrams to get a better and/or clearer understanding of the above-explanations.

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