`P ** q = p^2 - q^2`
`p $ q = p^2 + q^2`
`p @ q = pq + p + q`
`p Delta q` = Remainder of `p/q`
`p © q` = greatest integer less than or equal to `p/q`.
If `p = 15 and q = 25`, then the value of the expression `[(q**p)@(p $q)]` is:

Options

A. a.341200
B. b.341500
C. c.341250
D. d. none

Answer

To solve the expression \([(q ** p) @ (p $ q)]\) where \(p = 15\) and \(q = 25\), we need to break it down step by step according to the operations defined.

### Step 1: Calculate \(q ** p\)
The operation \(p ** q\) is defined as \(p^2 - q^2\). Therefore, we need to calculate \(q ** p\) as follows:

\[ q ** p = q^2 - p^2 \]
Substituting the values of \(p\) and \(q\):

\[ q ** p = 25^2 - 15^2 \]

Calculating the squares:

\[ 25^2 = 625 \quad \text{and} \quad 15^2 = 225 \]

Now, substituting these values back into the equation:

\[ q ** p = 625 - 225 = 400 \]

### Step 2: Calculate \(p $ q\)
The operation \(p $ q\) is defined as \(p^2 + q^2\). Therefore, we calculate:

\[ p $ q = p^2 + q^2 \]
Substituting the values of \(p\) and \(q\):

\[ p $ q = 15^2 + 25^2 \]

Calculating the squares again:

\[ 15^2 = 225 \quad \text{and} \quad 25^2 = 625 \]

Now, substituting these values back into the equation:

\[ p $ q = 225 + 625 = 850 \]

### Step 3: Calculate \((q ** p) @ (p $ q)\)
Now we need to calculate \((q ** p) @ (p $ q)\) which is defined as \(pq + p + q\). We already found \(q ** p = 400\) and \(p $ q = 850\), so we need to calculate:

\[ (q ** p) @ (p $ q) = 400 @ 850 \]

Substituting into the operation:

\[ 400 @ 850 = 400 \cdot 850 + 400 + 850 \]

Calculating \(400 \cdot 850\):

\[ 400 \cdot 850 = 340000 \]

Now adding \(400\) and \(850\):

\[ 400 + 850 = 1250 \]

Finally, adding these results together:

\[ 400 @ 850 = 340000 + 1250 = 341250 \]

### Final Answer
Thus, the value of the expression \([(q ** p) @ (p $ q)]\) is:

\[ \boxed{341250} \]