If $X$ and $Y$ are two sets such that $X$ has $40$ elements, $X \cup Y$ has $60$ elements and $X$ $\cap\, Y$ has $10$ elements, how many elements does $Y$ have?

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It is given that:

$n(X)=40, n(X \cup Y)=60, n(X \cap Y)=10$

We know that:

$n(X \cup Y)=n(X)+n(Y)-n(X \cap Y)$

$\therefore 60=40+n(Y)-10$

$\therefore n(Y)=60-(40-10)=30$

Thus, the set $Y$ has $30$ elements.

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