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(A) The question paper has $3$ parts,each with $4$ questions. We need to select $8$ questions in total,with at least $2$ from each part.Let $n_1, n_2,

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$396$

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(A) The question paper has $3$ parts,each with $4$ questions. We need to select $8$ questions in total,with at least $2$ from each part.Let $n_1, n_2, n_3$ be the number of questions selected from part $1, 2,$ and $3$ respectively.We have $n_1 + n_2 + n_3 = 8$,where $2 \le n_i \le 4$ for $i = 1, 2, 3$.The possible sets of $(n_1, n_2, n_3)$ are permutations of $(4, 2, 2)$ and $(3, 3, 2)$.Case $1$: $(4, 2, 2)$ in any order. There are $3$ such arrangements: $(4, 2, 2), (2, 4, 2), (2, 2, 4)$.Number of ways $= 3 	imes (^{4}C_4 	imes ^{4}C_2 	imes ^{4}C_2) = 3 	imes (1 	imes 6 	imes 6) = 3 	imes 36 = 108$.Case $2$: $(3, 3, 2)$ in any order. There are $3$ such arrangements: $(3, 3, 2), (3, 2, 3), (2, 3, 3)$.Number of ways $= 3 	imes (^{4}C_3 	imes ^{4}C_3 	imes ^{4}C_2) = 3 	imes (4 	imes 4 	imes 6) = 3 	imes 96 = 288$.Total ways $= 108 + 288 = 396$.

$A$ question paper has $3$ parts and each part contains $4$ questions. The number of different ways in which a candidate can answer $8$ questions choosing at least two from each part is

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