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It is a bit difficult to understand the question. But are you asking for the number of different forcing extensions by a given partial order? From the perspective of the universe that we are living in? Or from the outside?

Let me be more specific. Let $M$ be a countable transitive model of set theory. In $M$, let $P$ be Cohen forcing, i.e., the partial order of functions from a natural number to $2$, ordered by reverse inclusion.
Now, $P$ is a countable set, both in the real world and from the perspective of $M$.

In the real world, there are $2^{\aleph_0}$ different $P$-generic filters over $M$.
($2^{\aleph_0}$ is clearly an upper bound, but it can also be shown that there are $2^{\aleph_0}$ different ones.) Now, some of these filters might actually give the same generic extension, for example if one filter is just a finite modification of the other. But for each generic filter $G$ the extension $M[G]$ is countable (in the real world). Hence, if we have $2^{\aleph_0}$ different generic filter filters that each yield only a countable extension of $M$, there must be $2^{\aleph_0}$ pairwise different generic extensions of $M$.

Now, if you are interested in the number of extensions from the perspective of $M$, that question does not really make sense, since $M$ doesn't have access to the generic filters over $M$.
For each cardinal $\kappa$ of $M$, there is a forcing extension of $M$ by iterated Cohen forcing in which $2^{\aleph_0}$ is at least $\kappa$. Note that $M$ and this large generic extension have the same cardinals. And that forcing extension contains, as substructures, $\kappa$ many different generic extensions of $M$ by single Cohen reals by the same argument as above.

Since $\kappa$ was an arbitrary cardinal of $M$, this shows that it is impossible to gauge the number of generic extensions from inside the ground model.

I hope this does at least tell you how the question should be phrased.

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It is a bit difficult to understand the question. But are you asking for the number of different forcing extensions by a given partial order? From the perspective of the universe that we are living in? Or from the outside?

Let me be more specific. Let $M$ be a countable transitive model of set theory. In $M$, let $P$ be Cohen forcing, i.e., the partial order of functions from a natural number to $2$, ordered by reverse inclusion.
Now, $P$ is a countable set, both in the real world and from the perspective of $M$.

In the real world, there are $2^{\aleph_0}$ different $P$-generic filters over $M$.
($2^{\aleph_0}$ is clearly an upper bound, but it can also be shown that there are $2^{\aleph_0}$ different ones.) Now, some of these filters might actually give the same generic extension, for example if one filter is just a finite modification of the other. But for each generic filter $G$ the extension $M[G]$ is countable (in the real world). Hence, if we have $2^{\aleph_0}$ different generic filter that each yield only a countable extension of $M$, there must be $2^{\aleph_0}$ pairwise different generic extensions of $M$.

Now, if you are interested in the number of extensions from the perspective of $M$, that question does not really make sense, since $M$ doesn't have access to the generic filters over $M$.
For each cardinal $\kappa$ of $M$, there is a forcing extension of $M$ by iterated Cohen forcing in which $2^{\aleph_0}$ is at least $\kappa$. Note that $M$ and this large generic extension have the same cardinals. And that forcing extension contains, as substructures, $\kappa$ many different generic extensions of $M$ by single Cohen reals by the same argument as above.

Since $\kappa$ was an arbitrary cardinal of $M$, this shows that it is impossible to gauge the number of generic extensions from inside the ground model.

I hope this does at least tell you how the question should be phrased.