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Q.
$X, Y, Z$ are sets of all positive divisors of $10^{60}$, $20^{50}$ and $30^{40}$ respectively $n ( X \cup Y \cup Z )$ is
Permutations and Combinations
Solution:
$x =2 \times 5^{60} ; n ( x )=6 ! \times 5 !$
$n ( y )=10 ! \times 5 !, where y -2^{100} \times 5^{10}$
$z =2^{40} \times 3^{40} \times 5^{40}$
$n ( z )=41^{3}$
$n ( x \cap y )=6 ! \times 5 !$
$n ( x \cap z )=41^{2}= n ( z \cap x )$
$n ( x \cap y \cap z )=41^{2}$
$n ( x \cup y \cup z )= n ( x )+ n ( y )+ n ( z )$
$= n ( x \cap y )-( y \cap z )- n ( z \cap x )+ n ( x \cap y \cap z )$
$=61^{2}+101 \times 51+(41)^{3}-61 \times 51-41^{2}-41^{2}$
$=61(61-51)+41^{2}(41-1)+101 \times 51=73001$