Question 1196153
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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.  


<pre>
    (notice that his estimations do not disprove mine: they simply are {{{highlight(more_advanced)}}} estimations).
</pre>

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.



    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Thanks to tutor @MathTherapy for really nice job !


<H3>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- - - Updated version - - -</H3>

<pre>
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.


<U>ANSWER</U>.  The number of people in the orchestra is not lesser than 35 and not greater than 45.
</pre>

Solved.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;At the end, &nbsp;I have one more notice to a reader and to tutor @MathTherapy.



In the analysis, we absolutely correctly arrive to the conclusion that &nbsp;n(B ∩ P) &nbsp;is between &nbsp;0 &nbsp;and &nbsp;10.


But we have no reasons to reject the end-point possibilities  &nbsp;n(B ∩ P) = 0  &nbsp;or &nbsp;n(B ∩ P) = 10.


Therefore, my answer  &nbsp;&nbsp;35 <= n(orchestra) <= 45  &nbsp;&nbsp;is still different 

from  the @MathTherapy's answer  &nbsp;&nbsp;36 <= n(orchestra) <= 44.


<H3>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- - - My previous, now obsolete version - - - </H3>

<pre>
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.


<U>ANSWER</U>.  The number of people in the orchestra is not lesser than 31 and not greater than 45.
</pre>