We are given a Venn diagram with two sets, S and N, within a universal set U. The number of elements in S is 110, denoted as $n(S) = 110$. The number of elements in N is 104, denoted as $n(N) = 104$. The number of elements in the universal set U is 150, denoted as $n(U) = 150$. We need to find the number of elements in the intersection of S and N, which is denoted as $x$, or $n(S \cap N)$.

1. Problem Description

We are given a Venn diagram with two sets, S and N, within a universal set U.

The number of elements in S is 110, denoted as $n(S) = 110$.

The number of elements in N is 104, denoted as $n(N) = 104$.

The number of elements in the universal set U is 150, denoted as $n(U) = 150$.

We need to find the number of elements in the intersection of S and N, which is denoted as $x$, or $n(S \cap N)$.

2. Solution Steps

We know that

$n(S \cup N) = n(S) + n(N) - n(S \cap N)$.

We also know that the number of elements in the universal set is 150, which means $n(U) = 150$.

From the Venn diagram, we know that all elements of S and N are within U.

Therefore, $n(S \cup N) \le n(U)$.

Since $n(U) = 150$,

$n(S) = 110$, and

$n(N) = 104$,

$n(S \cup N) = n(S) + n(N) - n(S \cap N)$

$n(S \cup N) = 110 + 104 - x$

$n(S \cup N) = 214 - x$

Since $n(S \cup N) \le n(U)$, we have:

$214 - x \le 150$

$214 - 150 \le x$

$64 \le x$

Also, $x$ cannot be greater than $n(N)$ or $n(S)$.

$x \le n(S) = 110$ and $x \le n(N) = 104$. Therefore $x \le 104$.

We are given that $n(S \cup N)$ + the number of elements outside $S \cup N$ is equal to $n(U)$. Let $y$ be the number of elements outside $S \cup N$. Then

$n(S \cup N) + y = n(U)$

$214 - x + y = 150$

$y = x - 64$

Since $y \ge 0$, $x \ge 64$.

However, the diagram provides no information to determine a specific value for $x$. If we assume that all the elements in the universal set are in $S \cup N$, then $y=0$ and $x=64$.

However, without additional information, we can only conclude that $64 \le x \le 104$. But since the problem only asks for a value of x which is $n(S \cap N)$, we cannot determine the exact value. We can assume that $n(S \cup N)=n(U)=150$.

Then, $214-x = 150$, so $x=214-150 = 64$.

3. Final Answer

64