nicekikah

2021-06-13

Let $n\left(X\cdot Y\right)=24,n\left(X\cdot Z\right)=15$, and $n\left(Y\cdot Z\right)=40$. Find $n\left(X\cdot Y\cdot Z\right).$

Talisha

We know that: $n\left(X\cdot Y\right)=n\left(X\right)\cdot n\left(Y\right)$, $n\left(X\cdot Z\right)=n\left(X\right)\cdot n\left(Z\right)$, $n\left(Y\cdot Z\right)=n\left(Y\right)\cdot n\left(Z\right)$,
and
$x\left(X\cdot Y\cdot Z\right)=n\left(X\right)\cdot n\left(Y\right)\cdot n\left(Z\right)$
Therefore, $n\left(X\right)\cdot n\left(Y\right)=24,n\left(X\right)\cdot n\left(Z\right)=15$
and
$n\left(Y\right)\cdot n\left(Z\right)=40$
Multiplying the three equations, we get: ${\left(n\left(X\right)\right)}^{2}{\left(n\left(Y\right)\right)}^{2}{\left(n\left(Z\right)\right)}^{2}=24\cdot 15\cdot 40=14400$
Thus, ${\left(n\left(X\right)\cdot n\left(Y\right)\cdot n\left(Z\right)\right)}^{2}=14400\to n\left(X\right)\cdot n\left(Y\right)\cdot n\left(Z\right)=120$
Finally, using (*), $n\left(X\cdot Y\cdot Z\right)=120$

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