If P(77, 31) = x and C (77, 31) = y, then which one of the following is correct?

Options

A. x = y
B. 2x = y
C. 77x = 31 y
D. x `gt` y

Answer

To solve the problem, we need to understand the definitions of permutations and combinations.

1. **Understanding Permutations (P)**:
The number of ways to arrange 'r' objects from 'n' distinct objects is given by the formula:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
In this case, we have:
\[ P(77, 31) = \frac{77!}{(77-31)!} = \frac{77!}{46!} \]
Therefore, we can denote:
\[ x = P(77, 31) = \frac{77!}{46!} \]

2. **Understanding Combinations (C)**:
The number of ways to choose 'r' objects from 'n' distinct objects is given by the formula:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
In this case, we have:
\[ C(77, 31) = \frac{77!}{31!(77-31)!} = \frac{77!}{31! \cdot 46!} \]
Therefore, we can denote:
\[ y = C(77, 31) = \frac{77!}{31! \cdot 46!} \]

3. **Comparing x and y**:
Now we have:
\[ x = \frac{77!}{46!} \]
\[ y = \frac{77!}{31! \cdot 46!} \]

To compare \(x\) and \(y\), we can express \(x\) in terms of \(y\):
\[ x = y \cdot 31! \]

4. **Conclusion**:
Since \(31! > 1\), it follows that:
\[ x > y \]

Thus, the correct option is that \(x\) is greater than \(y\).

### Final Answer:
The correct option is: \(x > y\).