Question

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

Answer 1

Answer:

The coefficient of a²b³c is -720

Step-by-step explanation:

Given: $(2a-b+3c)^6$

Let 2a-b = x and 3c = y

$(x+y)^6$

General term of binomial expansion.

• $T_{r+1}=^nC_rx^{n-r}y^r$

where, n=6 , r=1 ( because exponent of c is 1)

$\Rightarrow ^6C_1x^{6-1}y^1$

$\Rightarrow 6(2a-b)^{5}3c$         $\because y=3c, x=2a-b$

$\Rightarrow 18(2a-b)^{5}c$  ----------(1)

Now, we simplify (2a-b)⁵

$T_{r+1}=^5C_r(2a)^{5-r}(-b)^r$

The exponent of b is 3 and a is 2 .

If we take r=3 will get exponent of b is 3 and a is 2

So, put r=3

$T_{4}=^5C_3(2a)^{2}(-b)^3=-40a^2b^3$

Substitute into equation (1)

$\Rightarrow 18\cdot -40a^2b^3c$

$\Rightarrow -720a^2b^3c$

Hence, The coefficient of a²b³c is -720