The half-life period of a radioactive element x is same as the mean life time of another radioactive element y. Initially, both of them have the same number of atoms. Then,
(a) x and y have the same decay rate initially
(b) x and y decay at the same rate always
(c) y will decay at a faster rate than x
(d) x will decay at a faster rate than y

Explanation

To solve the problem, we need to analyze the relationship between the half-life period of radioactive element X and the mean lifetime of radioactive element Y.

### Step-by-Step Solution:

1. **Understanding Half-Life and Mean Life**:
- The half-life (T_1/2) of a radioactive element is the time required for half of the radioactive atoms to decay.
- The mean lifetime (τ) of a radioactive element is the average time that a single atom will exist before decaying.

2. **Given Information**:
- We are told that the half-life of element X is the same as the mean lifetime of element Y.
- Mathematically, this can be expressed as:
\[ T_{1/2} = \tau_Y \]

3. **Formulas**:
- The relationship between half-life and decay constant (λ) is given by:
\[ T_{1/2} = \frac{\ln 2}{\lambda_X} \]
- The mean lifetime is related to the decay constant by:
\[ \tau_Y = \frac{1}{\lambda_Y} \]

4. **Setting Up the Equation**:
- Since \( T_{1/2} = \tau_Y \), we can equate the two expressions:
\[ \frac{\ln 2}{\lambda_X} = \frac{1}{\lambda_Y} \]

5. **Rearranging the Equation**:
- Rearranging gives:
\[ \lambda_Y = \frac{\lambda_X}{\ln 2} \]
- Since \( \ln 2 \approx 0.693 \), we can express this as:
\[ \lambda_Y \approx 1.4427 \lambda_X \]

6. **Comparing Decay Rates**:
- From the above relationship, we can see that:
\[ \lambda_Y > \lambda_X \]
- This means that element Y has a higher decay constant than element X, indicating that Y decays faster than X.

7. **Conclusion**:
- Therefore, the correct answer is that element Y will decay at a faster rate than element X.

### Final Answer:
(c) y will decay at a faster rate than x.