First slide

Binomial theorem for positive integral Index

Question

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

Moderate

A

6
B

60
C

61
D

None of these

Solution

In this case, we write
$\begin{aligned} a^{3} b^{4} c^{5} = (ab)^{x} (bc)^{y} (ca)^{z} \\ \Rightarrow a^{3} b^{4} c^{5} = a^{z + x} \cdot b^{x + y} \cdot c^{y + z} \\ \Rightarrow z + x = 3, x + y = 4, y + z = 5 . . . . . (1) \end{aligned}$
On adding these equations, we get
$∴ \begin{aligned} 2 (x + y + z) = 12 \\ x + y + z = 6 \end{aligned} . . . . . . . (2)$
On solving Eqs. (i) and (ii), we get
x=1,Y =3 ,Z =2
Therefore, the coefficient of $a^{3} b^{4} c^{5}$ in the expansion of
(bc+ ca+ ab)⁶
The coefficient of (ab)¹(bc)³(ca)² in the expansion of
(bc+ca+ab)⁶
$= \frac{6!}{1! 3! 2!} = 60$

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