<p>The coefficient of <math>a^3b^4c^5</math> in the expansion of <math>(bc + ca + ab)^6</math> is</p>
Step-by-Step Solution
Key Concept: Use the multinomial theorem and match the powers of a, b, and c in the expansion to find the required multinomial coefficient.
<p><strong>Solution:</strong></p><p>The general term in the expansion of <math>(bc + ca + ab)^6</math> is:</p><p><math>\frac{6!}{p! q! r!} (bc)^p (ca)^q (ab)^r</math></p><p>where <math>p + q + r = 6</math></p><p><math>= \frac{6!}{p! q! r!} b^p c^p c^q a^q a^r b^r = \frac{6!}{p! q! r!} a^{q+r} b^{p+r} c^{p+q}</math></p><p>For the coefficient of <math>a^3b^4c^5</math>, we need:</p><p><math>q + r = 3, p + r = 4, p + q = 5</math></p><p>Solving: <math>p = 2, q = 3, r = 1</math></p><p><math>\text{Coefficient} = \frac{6!}{2! \times 3! \times 1!} = \frac{720}{2 \times 6 \times 1} = 60</math></p>
Correct Answer: b