The coefficient of `x^(5)` in the expansion of `(2 - x + 3x^(2))^(6)` , is

Options

A. 3
B. 4
C. 5
D. none of these

Answer

a

Explanation

The gerneral term in the expansion of
`(2 - x+ 3x^(2))^(3)` is
`(6!)/(r!s!t!)2^(r) (-x)^(s) (3x^(2))^(t)` , where r + s + t = 6.
`(6!)/(r!s!t!)2^(r) xx(-1)^(s)xx 3^(t)xxx^(s + 2t)`, where r + s + t = 6
For the coefficient of `x^(5)` , we must have s + 2t = 5
But , r + s + t = 6
`therefore s = 5 - 2t and (r =1 + t` , where 0 `le r, s, tle 6`
Now,
t = 0 `rArr r = 1 , s = 5`
`t = 1 rArr r = 2 , s = 3`
`t = 2 rArr r = 3 , s = 1`
Hence, there are 3 terms containing `x^(5)`