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?
<pre> *[illustration ADC_Problem_1196153_01.png].
which leads to:
 *[illustration ADC_Problem_1196153_111.png].
Although not stated, the following is based on the premise that each member can play at least 1 instrument. 

Let S, B, and P be the number of persons who can play stringed, brass, and percussion instruments, respectively
Also, let B&P be the number that can play both brass and percussion instruments

Then: Number of persons who can play stringed instrments, ONLY: 22 - 8 - 4 = 10
      Number of persons who can play brass instruments, ONLY: 21 - 8 - B&P = 13 - B&P
      Number of persons who can play percussion instrments, ONLY: 14 - 4 - B&P = 10 - B&P
      Total number in orchestra would then be: 10 + 8 + 4 + (13 - B&P) + B&P + (10 - B&P) = 45 - B&P

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
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. 

If B&P is 1, then number in orchestra = 45 - 1, or 10 + 8 + 4 + 13 - 1 + 1 + 10 - 1 = 44
If B&P is 9, then number in orchestra = 45 - 9, or 10 + 8 + 4 + 13 - 9 + 9 + 10 - 9 = 36

So, the <font color = red><font size = 4><b>MAXIMUM</font></font></b> number that can be in the orchestra is <font color = red><font size = 4><b>44</font></font></b> and the <font color = red><font size = 4><b>MINIMUM</font></font></b> number is <font color = red><font size = 4><b>36</font></font></b>.

Refer to the above Venn Diagrams to get a better and/or clearer understanding of the above-explanations.</pre>