For any natural number p and q
(i) `p # q = p^3 + q^3 + 3 and p ** q = p^2 + q^2 + 2` and `p $ q = |p - q|`
(ii) Max (p,q) = Maximum of (p,q) and Min (p and q) = Minimum of (p,q)
The value of `[(4 # 5) $ (14 ** 15)]` is :

Options

A. a. `-196`
B. b. `231`
C. c. `-225`
D. d. `229`

Answer

To solve the problem step by step, we will follow the operations defined for the symbols #, **, and $.

### Step 1: Calculate \( 4 \# 5 \)
According to the problem, the operation \( p \# q \) is defined as:
\[ p \# q = p^3 + q^3 + 3 \]
Substituting \( p = 4 \) and \( q = 5 \):
\[ 4 \# 5 = 4^3 + 5^3 + 3 \]
Calculating \( 4^3 \) and \( 5^3 \):
\[ 4^3 = 64 \quad \text{and} \quad 5^3 = 125 \]
Now substituting these values:
\[ 4 \# 5 = 64 + 125 + 3 = 192 \]

### Step 2: Calculate \( 14 ** 15 \)
The operation \( p ** q \) is defined as:
\[ p ** q = p^2 + q^2 + 2 \]
Substituting \( p = 14 \) and \( q = 15 \):
\[ 14 ** 15 = 14^2 + 15^2 + 2 \]
Calculating \( 14^2 \) and \( 15^2 \):
\[ 14^2 = 196 \quad \text{and} \quad 15^2 = 225 \]
Now substituting these values:
\[ 14 ** 15 = 196 + 225 + 2 = 423 \]

### Step 3: Calculate \( (4 \# 5) $ (14 ** 15) \)
Now we need to evaluate \( 192 $ 423 \). The operation \( p $ q \) is defined as:
\[ p $ q = |p - q| \]
Substituting \( p = 192 \) and \( q = 423 \):
\[ 192 $ 423 = |192 - 423| \]
Calculating \( 192 - 423 \):
\[ 192 - 423 = -231 \]
Taking the absolute value:
\[ | -231 | = 231 \]

### Final Answer
Thus, the value of \( [(4 \# 5) $ (14 ** 15)] \) is:
\[ \boxed{231} \]