Find the coefficient of $a^{2} b^{3} c^{4} d$ in the expansion of ${(a - b - c + d)}^{10}$ be $k$ . Find sum of digits of $| k |$ ?

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Solution

$(a - b - c + d)^{10}$
Coefficient of $(a)^{r_{1}} \times {(b)}^{r_{2}} \times {(c)}^{r_{3}} \times {(d)}^{r_{4}}$ is
$\frac{10!}{r_{1}! . r_{2}! . r_{3}! . r_{4}!} 1^{r_{1}} {(- 1)}^{r_{2}} {(- 1)}^{r_{3}} {(1)}^{r_{4}}$
$= \frac{10! {(- 1)}^{3} {(- 1)}^{4}}{2! \times 3! \times 4! \times 1!} = - 12600$

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