Correct Answer: Option C\n\nHere is a step-by-step solution:\n\n**Step 1: Understand the initial state of the class.**\n\n* Number of students in the class initially = $30$\n* Average score of the class initially = $X$\n* Total score of the class initially = Number of students $\\times$ Average score $= 30X$\n\n**Step 2: Analyze the first scenario (5 new students join).**\n\n* Number of new students = $5$\n* Average score of these $5$ new students = $M$\n* Total score contributed by these $5$ new students = $5M$\n* Given that $M = 60$, the total score contributed by these $5$ new students is $5 \\times 60 = 300$.\n\n* New total number of students in the class = $30 + 5 = 35$\n* New average score of the class = $X - 2$\n* New total score of the class = $35(X - 2)$\n\n* The total score of the new class is the sum of the initial total score and the total score of the new students:\n \[\n 30X + 5M = 35(X - 2)\n \]\n* Substitute the value of $M$ into the equation:\n \[\n 30X + 5(60) = 35X - 70\n \]\n \[\n 30X + 300 = 35X - 70\n \]\n* Rearrange the terms to solve for $X$:\n \[\n 300 + 70 = 35X - 30X\n \]\n \[\n 370 = 5X\n \]\n \[\n X = \\frac{370}{5}\n \]\n \[\n X = 74\n \]\n So, the initial average score of the class ($X$) is $74$.\n\n**Step 3: Analyze the second scenario (5 original students leave).**\n\n* Number of students leaving = $5$\n* Average score of these $5$ leaving students = $N$\n* Total score contributed by these $5$ leaving students = $5N$\n\n* Remaining number of students in the class = $30 - 5 = 25$\n* New average score of the remaining students = $X + 1$\n* New total score of the remaining students = $25(X + 1)$\n\n* The total score of the remaining students is the initial total score minus the total score of the students who left:\n \[\n 30X - 5N = 25(X + 1)\n \]\n\n**Step 4: Solve for $N$ using the value of $X$.**\n\n* We found $X = 74$ in Step 2. Substitute this value into the equation from Step 3:\n \[\n 30(74) - 5N = 25(74 + 1)\n \]\n \[\n 2220 - 5N = 25(75)\n \]\n \[\n 2220 - 5N = 1875\n \]\n* Rearrange the terms to solve for $N$:\n \[\n 2220 - 1875 = 5N\n \]\n \[\n 345 = 5N\n \]\n \[\n N = \\frac{345}{5}\n \]\n \[\n N = 69\n \]\n\nThus, the average score of the $5$ students who left ($N$) is $69$.\n\nThe final answer is $\\boxed{69}$.