Question

The coefficient of $a^{10}{b}^7{c}^3$ in the expansion of $(bc+ca+ab)10$ is

• A. $30$
• B. $60$
• C. $120$
• D. $240$

Answer 1

The correct option is C $120$

The general term in the expansion of $(bc+ca+ab{)}^{10}$ is

$\frac{10!}{r!s!t!}\left(bc{)}^r\right(ca{)}^s(ab{)}^t=\frac{10!}{r!s!t!}{a}^{s+t}{b}^{r+t}{c}^{r+s}$

where $r+s+t=10$

For the coefficient of $a^{10}{b}^7{c}^3$ we take $s+t=10,r+t=7,r+s=3$

Since $r+s+t=10$ we get $r=0,s=3,t=7$

Thus, the coefficient of $a^{10}{b}^7{c}^3$ in the expansion of $(bc+ca+ab{)}^{10}$ is

$=\frac{10!}{0!3!7!}$

$=\frac{10\times 9\times 8}{1\times 3\times 2}=120$