<p>The number of dissimilar terms in the expansion of \((a + 2b + 3c)^8\) is</p>
Step-by-Step Solution
Key Concept: In the multinomial expansion of (a + 2b + 3c)^8, each distinct term corresponds to a unique way of distributing 8 among three variables. Use the stars and bars formula: the number of non-negative integer solutions to p + q + r = 8 is C(8+3-1, 3-1) = C(10, 2).
<p><strong>Step 1:</strong> The general term in the expansion of (a + 2b + 3c)^8 is of the form a^p(2b)^q(3c)^r where p + q + r = 8 and p, q, r ≥ 0.</p><p><strong>Step 2:</strong> Each dissimilar term corresponds to a unique triple (p, q, r). The number of non-negative integer solutions to p + q + r = 8 is given by the stars and bars formula: C(n+k-1, k-1) where n = 8 (total power) and k = 3 (number of variables).</p><p><strong>Step 3:</strong> Number of solutions = C(8+3-1, 3-1) = C(10, 2) = 10×9/2 = 45</p><p>∴ Answer: C (45)</p>
Correct Answer: C