If (1 − x^3 )^n = Σ_(r=0) ^n a_r x^r (1 − x)^(3n−2r) , then the value of a_r , where n ∈ N is (1)^n C_r ∙3^r (2)^n C_(3r) (3)^n C_(r−1) 2^(r−1) (4)^n C


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Question Number 23700 by Tinkutara last updated on 04/Nov/17

$If {(1 - x^{3})}^{n} = \underset{r = 0}{\sum^{n}} a_{r} x^{r} {(1 - x)}^{3 n - 2 r}, then$ $the value of a_{r}, where n \in N is$ $(1)^{n} C_{r} \cdot 3^{r}$ $(2)^{n} C_{3 r}$ $(3)^{n} C_{r - 1} 2^{r - 1}$ $(4)^{n} C_{r} 2^{r}$
Commented by ajfour last updated on 04/Nov/17

$I f n = 1$ $1 - x^{3} = a_{0} {(1 - x)}^{3} + a_{1} x (1 - x)$ $\Rightarrow a_{0} = 1 a n d a_{1} = 3$ $i f w e c h e c k o p t i o n s w i t h n = 1$ $(1) g i v e s a_{0} = 1, a_{1} = 3$ $(2) a_{0} = 1, a_{1} =!$ $(3) a_{0} =!, a_{1} = 1$ $(4) a_{0} = 1, a_{1} = 2$ $S o o n l y o p t i o n (1) q u a l i f i e s . .$
Commented by Tinkutara last updated on 05/Nov/17

$Not any proof ?$
Commented by ajfour last updated on 05/Nov/17

$i w i l l t r y . .$
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