In University $X$, 60 candidates applied for a PhD course with the following qualifications:
- 30 candidates have a master's degree.
- $\(\mathbf{2 0}\)$ candidates have assistantship experience of at least 3 years.
- 10 candidates have assistantship experience of less than 3 years and do not have a master's degree.

We need to find how many candidates have both:
- A master's degree and
- Assistantship experience of at least 3 years.

Step 1: Define the Groups
Let's categorize the candidates based on their qualifications:
1. Master's Degree (M): 30 candidates.
2. Assistantship $\(\geq \mathbf{3}\)$ Years (A): 20 candidates.
3. Assistantship < 3 Years and No Master's Degree: 10 candidates.

Step 2: Determine the Unknown
We need to find the number of candidates with both $\mathbf{M}$ and $\mathbf{A}$, denoted as $\(M \cap A\)$.

Step 3: Use the Principle of Inclusion-Exclusion
The total number of candidates is 60 . We can express this as:

$$
\(\text { Total }=\text { Only } \mathrm{M}+\text { Only } \mathrm{A}+\text { Both } \mathrm{M} \text { and } \mathrm{A}+\text { Neither } \mathrm{M} \text { nor } \mathrm{A}\)
$$



From the problem:
- Neither M nor A corresponds to candidates with < 3 years assistantship and no master's degree, which is given as 10 .
- Only A is not directly given, but we can find it.

However, let's simplify using a table or Venn diagram approach.
Step 4: Break Down the Groups
1. Candidates with < 3 Years Assistantship and No Master's Degree: 10 (given).
2. Remaining Candidates: $60-10=50$ have either:
- A master's degree,
- $\\(geq 3\)$ years assistantship, or
- Both.

Now, apply the inclusion-exclusion principle to these 50 candidates:

$$
\(\mathrm{M}+\mathrm{A}-\mathrm{M} \cap \mathrm{~A}=50\)
$$


Substitute the known values:

$$
\(\begin{gathered}
30+20-\mathrm{M} \cap \mathrm{~A}=50 \\
50-\mathrm{M} \cap \mathrm{~A}=50 \\
\mathrm{M} \cap \mathrm{~A}=0
\end{gathered}\)
$$


Verification
- Only Master's Degree: $30-0=30$
- Only $\(\geq 3\)$ Years Assistantship: $\(20-0=20\)$
- Both: 0
- Neither (given): 10

Total: $\(30+20+0+10=60\)$ (matches the total candidates).
Conclusion
No candidates have both a master's degree and at least 3 years of assistantship experience.
Final Answer
\(\boxed\{0\}\)