If \( a_{i}, b_{i} \in \mathbf{N} \) for \( i=1,2,3 \) then the coe...

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If \( a_{i}, b_{i} \in \mathbf{N} \) for \( i=1,2,3 \) then the coefficient of \( x \) in the determinant
\[
\left|\begin{array}{lll}
(1+x)^{a_{1} b_{1}} & (1+x)^{a_{1} b_{2}} & (1+x)^{a_{1} b_{3}} \\
(1+x)^{a_{2} b_{1}} & (1+x)^{a_{2} b_{2}} & (1+x)^{a_{2} b_{3}} \\
(1+x)^{a_{3} b_{1}} & (1+x)^{a_{3} b_{2}} & (1+x)^{a_{3} b_{3}}
\end{array}\right|
\]
(a) 0
(b) \( a_{1}+a_{2}+a_{3} \)
(c) independent of \( a_{i}, s b_{i}, s \)
(d) \( b_{1}+b_{2}+b_{3} \)
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