Consider a prototypical fullerene, `C_(60)`
Let,` a` = Number of `5`-membered rings: `b` = Number of `6`-membered rings
`c` = Number of `pi` bonds in `C_(60)`
Find the value of `(3a-2b+c)`

Answer

To solve the problem, we need to determine the values of \(a\), \(b\), and \(c\) for the fullerene \(C_{60}\), and then calculate the expression \(3a - 2b + c\).

### Step 1: Identify the number of 5-membered rings (\(a\))
In the structure of \(C_{60}\), there are 12 five-membered rings. Therefore, we have:
\[ a = 12 \]

**Hint for Step 1:** Look at the structure of the fullerene to count the number of pentagons (5-membered rings).

### Step 2: Identify the number of 6-membered rings (\(b\))
In \(C_{60}\), there are 20 six-membered rings. Thus, we have:
\[ b = 20 \]

**Hint for Step 2:** Count the hexagons (6-membered rings) in the fullerene structure.

### Step 3: Identify the number of pi bonds (\(c\))
Each bond in \(C_{60}\) contributes to the total number of pi bonds. In \(C_{60}\), there are 30 pi bonds. Therefore, we have:
\[ c = 30 \]

**Hint for Step 3:** Remember that each double bond contributes one pi bond, and count the total number of pi bonds in the structure.

### Step 4: Substitute the values into the expression \(3a - 2b + c\)
Now we can substitute the values of \(a\), \(b\), and \(c\) into the expression:
\[ 3a - 2b + c = 3(12) - 2(20) + 30 \]

### Step 5: Calculate the expression
Now, let's perform the calculations:
\[ = 36 - 40 + 30 \]
\[ = 36 - 40 + 30 = 26 \]

### Final Answer
Thus, the value of \(3a - 2b + c\) is:
\[ \boxed{26} \]