Q70 / 155 · WBJEE 2024 · Mathematics
The coefficient of $a^{10} b^7 c^3$ in the expansion of $(b c+c a+a b)^{10}$ is
- A140
- B150
- Correct answerC120
- D160
Solution
Hint : \((b c+c a+a b)^{10}=\sum \frac{10}{\left\lfloor n_1\right\lfloor n_2 \lfloor n_3}(b c)^{n_1}(c a)^{n_2}(a b)^{n_3}=\sum \frac{10}{\left\lfloor n_1\right \lfloor n_2 \lfloor n_3} b^{n_1+n_3} c^{n_1+n_2} a^{n_2+n_3}\)
\(\begin{array}{l|l}\mathrm{n}_1+\mathrm{n}_3=7 & \mathrm{n}_1+\mathrm{n}_2+\mathrm{n}_3=10 \\ \mathrm{n}_1+\mathrm{n}_2=3 & \mathrm{n}_1=0, \mathrm{n}_2=3 \\ \mathrm{n}_2+\mathrm{n}_3=10 & \mathrm{n}_3=7\end{array}\)
Coefficient of \(a^{10} b^7 c^3=\frac{\lfloor 10}{\lfloor{3} \lfloor{7}}=\frac{10 \times 9 \times 8}{3 \times 2}=10 \times 3 \times 4=120\)
Hence, the correct answer is option: 'C'.