Question

Combinatorics: what is the coefficient of (a^2)(b^3)(c) in (2a - b + 3c)^6?

Answer 1

Hence, the coefficient of a²b³c = -720.

Explanation:

As from the question,

The general formula to find the coefficient is given by Binomial theorem:

That is, the coefficient of
`x^(\alpha)\cdot y^(\beta)\cdot z^(\gamma)` in (x + y + z)ⁿ is given by:

`(n!)/(\alpha ! \cdot \beta ! \cdot \gamma !) (x)^(\alpha) \cdot (y)^(\beta) \cdot (z)^(\gamma)`

Now,

From the question we have

`(2a-b+3c)^(6)` having n = 6

∴ x = 2a

y = -b

z = 3c

Now,

The coefficient of a²b³c, that is

α = 2

β = 3

γ = 1

Therefore the coefficient of a²b³c =

`= (6!)/(2 ! \cdot 3 ! \cdot 1 !) (2a)^(2) \cdot (-b)^(3) \cdot (3c)^(1)`

`= (6!)/(2 ! \cdot 3 ! \cdot 1 !) 4(a)^(2) \cdot (-b)^(3) \cdot (3c)`

= -720 a²b³c

Hence, the coefficient of a²b³c = -720.