p, q and r are three non-negative integers such that `p+q+r=10`. The maximum value of `pq+qr+pr+pqr` is

Options

A. `ge40andlt50`
B. `ge50andlt60`
C. `ge60andlt70`
D. `ge70andlt80`

Answer

To find the maximum value of \( pq + qr + pr + pqr \) given that \( p + q + r = 10 \) and \( p, q, r \) are non-negative integers, we can follow these steps:

### Step 1: Understand the constraint
We know that \( p + q + r = 10 \). This means that the sum of the three integers \( p, q, \) and \( r \) must equal 10.

### Step 2: Explore possible distributions of values
To maximize the expression \( pq + qr + pr + pqr \), we should consider how to distribute the total of 10 among \( p, q, \) and \( r \). Since we want to maximize the product terms, we should try to keep the values of \( p, q, \) and \( r \) as close to each other as possible.

### Step 3: Consider cases
We can divide 10 into three parts in different ways. The most balanced way to do this is to have two numbers equal and one number slightly larger.

1. **Case 1**: \( p = 3, q = 3, r = 4 \)
- Calculate \( pq + qr + pr + pqr \):
\[ pq = 3 \times 3 = 9 \]
\[ qr = 3 \times 4 = 12 \]
\[ pr = 3 \times 4 = 12 \]
\[ pqr = 3 \times 3 \times 4 = 36 \]
\[ \text{Total} = 9 + 12 + 12 + 36 = 69 \]

2. **Case 2**: \( p = 4, q = 3, r = 3 \)
- This will yield the same calculation as Case 1:
\[ pq = 4 \times 3 = 12 \]
\[ qr = 3 \times 3 = 9 \]
\[ pr = 4 \times 3 = 12 \]
\[ pqr = 4 \times 3 \times 3 = 36 \]
\[ \text{Total} = 12 + 9 + 12 + 36 = 69 \]

3. **Case 3**: \( p = 2, q = 4, r = 4 \)
- Calculate \( pq + qr + pr + pqr \):
\[ pq = 2 \times 4 = 8 \]
\[ qr = 4 \times 4 = 16 \]
\[ pr = 2 \times 4 = 8 \]
\[ pqr = 2 \times 4 \times 4 = 32 \]
\[ \text{Total} = 8 + 16 + 8 + 32 = 64 \]

### Step 4: Compare results
From the calculations, we see that the maximum value occurs in the first two cases, both yielding a total of 69.

### Conclusion
The maximum value of \( pq + qr + pr + pqr \) given \( p + q + r = 10 \) is:
\[ \boxed{69} \]