Question

Find the coefficient of $a^3{b}^4{c}^5$ int he expansion of ${(bc+ca+ab)}^6$ is

• A. $0$
• B. $30$
• C. $50$
• D. none of the above

Answer 1

Answer: (D)

none of the above

Using above result,
$(bc+ca+ab{)}^6=\sum \frac{6!\left(ab{)}^{{\alpha }_1}\right(bc{)}^{{\alpha }_2}(ca{)}^{{\alpha }_3}}{{\alpha }_1!{\alpha }_2!{\alpha }_3!}=\sum \frac{6!\left(a{)}^{{\alpha }_1+{\alpha }_3}\right(b{)}^{{\alpha }_1+{\alpha }_2}(c{)}^{{\alpha }_2+{\alpha }_3}}{{\alpha }_1!{\alpha }_2!{\alpha }_3!}$
Thus for coefficient of $a^3{b}^4{c}^5$ compare it with the above equation
$\Rightarrow {\alpha }_1+{\alpha }_3=3,{\alpha }_1+{\alpha }_2=4,{\alpha }_2+{\alpha }_3=5$
Also ${\alpha }_1+{\alpha }_2+{\alpha }_3=6$
Solving above we get, ${\alpha }_1=1,{\alpha }_2=3,{\alpha }_3=2$
Hence required coefficient $=\frac{6!}{1!2!3!}=\frac{720}{2\times 6}=60$