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?
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52858)   (Show Source):
You can put this solution on YOUR website! .
In an orchestra, 22 people can play stringed instruments, 21 can play brass,
and 14 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?
~~~~~~~~~~~~~
Tutor @MathTherapy in his post provided more advanced analysis,
which gave better estimations than my previous solution.
(notice that his estimations do not disprove mine: they simply are  estimations).
After reading his post, I got understanding where my solution can be improved,
so I place here new version, which gives the answer close to that of the @MathTherapy post.
Thus, below you see my improved solution, and after it, I still keep my previous, now obsolete version,
so a reader can compare and make conclusions for himself (or herself) - it might be educational.
I could delete my solution, at all - but still think that it can be useful for somebody,
who wants to see the solution to this problem based on Inclusive-Exclusive principle.
Thanks to tutor @MathTherapy for really nice job !
- - - Updated version - - -
We have a universal set U of all people of the orchestra, and three its subsets
S (stringed instruments) of 22 persons
B (brass) of 21 persons
P (percussion) of 14 persons.
We have information about intersections
n(S ∩ B) = 8;
n(S ∩ P) = 4,
n(S ∩ B ∩ P) = 0.
Write the inclusion-exclusion principle formula
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) =
= 22 + 21 + 14 - 8 - 4 - n(B ∩ P) + 0,
or
n(orchestra) = 45 - n(B ∩ P).
In this equality, the value of the term n(B ∩ P) is not given and is not known.
We only know that n(B ∩ P) is not greater than n(B) - n(B ∩ S) = 21-8 = 13 <<<---=== my corrections start here
and not greater than n(P) - n(S ∩ P) = 14-4 = 10.
So, n(B ∩ P) is between 0 and 10 (inclusive).
Therefore, n(orchestra) is not greater than 45 and not lesser than 45-10 = 35.
ANSWER. The number of people in the orchestra is not lesser than 35 and not greater than 45.
Solved.
At the end, I have one more notice to a reader and to tutor @MathTherapy.
In the analysis, we absolutely correctly arrive to the conclusion that n(B ∩ P) is between 0 and 10.
But we have no reasons to reject the end-point possibilities n(B ∩ P) = 0 or n(B ∩ P) = 10.
Therefore, my answer 35 <= n(orchestra) <= 45 is still different
from the @MathTherapy's answer 36 <= n(orchestra) <= 44.
- - - My previous, now obsolete version - - -
We have a universal set U of all people of the orchestra, and three its subsets
S (stringed instruments) of 22 persons
B (brass) of 21 persons
P (percussion) of 14 persons.
We have information about intersections
n(S ∩ B) = 8;
n(S ∩ P) = 4,
n(S ∩ B ∩ P) = 0.
Write the inclusion-exclusion principle formula
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) =
= 22 + 21 + 14 - 8 - 4 - n(B ∩ P) + 0,
or
n(orchestra) = 45 - n(B ∩ P).
In this equality, the value of the term n(B ∩ P) is not given and is not known.
We only know that n(B ∩ P) is not greater than the minimum min(21,14) = 14,
so we know that n(B ∩ P) is between 0 and 14.
Therefore, n(orchestra) is not greater than 45 and not lesser than 45-14 = 31.
ANSWER. The number of people in the orchestra is not lesser than 31 and not greater than 45.
Answer by MathTherapy(10556)  (Show Source):
You can put this solution on YOUR website! 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?
 .
which leads to:
 .
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 MAXIMUM number that can be in the orchestra is 44 and the MINIMUM number is 36.
Refer to the above Venn Diagrams to get a better and/or clearer understanding of the above-explanations.
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