To find the coefficient of t^24 in the expression (1 - t^2)^12 (1 - t^12) (1 - t^24), we need to expand the expression using the binomial theorem and then identify the term with t^24.

Expansion using the Binomial Theorem:
The binomial theorem states that for any positive integers n and k, the kth term in the expansion of (a + b)^n can be written as C(n, k) * a^(n-k) * b^k, where C(n, k) represents the binomial coefficient.

In this case, we can expand (1 - t^2)^12 and (1 - t^12) using the binomial theorem.

Expanding (1 - t^2)^12:
(1 - t^2)^12 = C(12, 0) * 1^12 * (-t^2)^0 + C(12, 1) * 1^11 * (-t^2)^1 + C(12, 2) * 1^10 * (-t^2)^2 + ... + C(12, 12) * 1^0 * (-t^2)^12

Simplifying the expression:
(1 - t^2)^12 = 1 - 12t^2 + 66t^4 - 220t^6 + 495t^8 - 792t^10 + 924t^12 - 792t^14 + 495t^16 - 220t^18 + 66t^20 - 12t^22 + t^24

Expanding (1 - t^12):
(1 - t^12) = 1 - t^12

Now, let's multiply the above two expressions with (1 - t^24):

(1 - t^2)^12 * (1 - t^12) * (1 - t^24) = (1 - t^2)^12 * (1 - t^12)
= (1 - t^2)^12 * 1 - t^12
= (1 - t^2)^12 - (1 - t^2)^12 * t^12

The coefficient of t^24 will be the coefficient of t^24 in (1 - t^2)^12 minus the coefficient of t^12 in (1 - t^2)^12 multiplied by t^12.

Coefficient of t^24:
From the expansion of (1 - t^2)^12, we can see that the term with t^24 is 1*(-12t^22) = -12t^22.

Coefficient of t^12:
From the expansion of (1 - t^2)^12, we can see that the term with t^12 is 66t^12.

Therefore, the coefficient of t^24 in (1 - t^2)^12 * (1 - t^12) * (1 - t^24) is -12 (coefficient of t^22) - 66t^12 (coefficient of t^12) = -12 - 66 = -78.

Correct answer: d) 12C6