If `sum_(r=0)^(n) (pr+2).""^(n)C_(r)=(25)(64)` where `n, p in N`, then (a) p=3 (b) p=4 (c) n=7 (d) n=6

Options

A. `p = 3`
B. `p = 4`
C. `n = 7`
D. `n = 6`

Explanation

To solve the problem, we need to evaluate the expression given in the summation and find the values of \( p \) and \( n \) such that:

\[ \sum_{r=0}^{n} (pr + 2) \binom{n}{r} = 25 \times 64 \]

### Step 1: Rewrite the Summation
Let:
\[ S = \sum_{r=0}^{n} (pr + 2) \binom{n}{r} \]

### Step 2: Split the Summation
We can split the summation into two parts:
\[ S = \sum_{r=0}^{n} pr \binom{n}{r} + \sum_{r=0}^{n} 2 \binom{n}{r} \]

### Step 3: Evaluate Each Part
1. The second part can be simplified using the binomial theorem:
\[ \sum_{r=0}^{n} \binom{n}{r} = 2^n \]
Thus,
\[ \sum_{r=0}^{n} 2 \binom{n}{r} = 2 \cdot 2^n = 2^{n+1} \]

2. For the first part, we use the identity:
\[ \sum_{r=0}^{n} r \binom{n}{r} = n \cdot 2^{n-1} \]
Therefore,
\[ \sum_{r=0}^{n} pr \binom{n}{r} = p \cdot n \cdot 2^{n-1} \]

### Step 4: Combine the Results
Combining both parts, we have:
\[ S = p \cdot n \cdot 2^{n-1} + 2^{n+1} \]

### Step 5: Set the Equation
We know from the problem statement that:
\[ S = 25 \times 64 \]
Calculating \( 25 \times 64 \):
\[ 25 \times 64 = 1600 \]
So we set up the equation:
\[ p \cdot n \cdot 2^{n-1} + 2^{n+1} = 1600 \]

### Step 6: Factor Out \( 2^{n-1} \)
Factoring out \( 2^{n-1} \):
\[ 2^{n-1}(pn + 4) = 1600 \]

### Step 7: Express \( 1600 \) in Terms of Powers of 2
We can express \( 1600 \) as:
\[ 1600 = 2^6 \cdot 25 = 2^6 \cdot 5^2 \]
Thus, we have:
\[ 2^{n-1}(pn + 4) = 2^6 \cdot 5^2 \]

### Step 8: Equate the Powers of 2
From the equation, we can equate the powers of 2:
\[ n - 1 = 6 \implies n = 7 \]

### Step 9: Substitute \( n \) Back to Find \( p \)
Now substituting \( n = 7 \) back into the equation:
\[ 2^6(p \cdot 7 + 4) = 1600 \]
Dividing both sides by \( 2^6 \):
\[ p \cdot 7 + 4 = 25 \]
Solving for \( p \):
\[ p \cdot 7 = 25 - 4 = 21 \implies p = \frac{21}{7} = 3 \]

### Conclusion
The values we found are:
- \( p = 3 \)
- \( n = 7 \)

### Final Answer
Thus, the correct options are:
- (a) \( p = 3 \)
- (c) \( n = 7 \)