N-Z=X+Y-Both
Both= X+Y-N+Z

Both/N= (X+Y-N+Z)/N

Answer: B

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make 2x2 matrix

---B-----NB---total
H--(y-(n-x)-z)---(n-x)-z---y
NH--(n-y)-z---z----n-y
total--x----n-x---n
fraction
(y-(n-x)-z)/n
or say
x+y+z-n/n ;
\(\frac{X+Y+Z}{N} - 1\)
IMO B

If there are two categories A and B, we have the formula Total = A + B - Both + Neither. To go into detail, the A category includes "A only" and "Both A and B". The B category includes "B only" and "Both A and B". So we double-counted "Both" if we just add everything up, hence we take it away once from the total. Here we want to find both,

Both = A + B + Neither - Total = X + Y + Z - N.
Finally, the fraction out of the total of N would be \(\frac{{X + Y + Z - N} }{ N} = \frac{{X + Y + Z} }{ N} - 1\). The answer would be B.
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N = X + Y - Both +Z
Both = X + Y + Z - N

Fraction of students who took both the courses = (X + Y + Z - N)/N = (X+Y+Z)/N - 1

IMO B
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