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(D) The question paper has three parts $A, B, C$ with $4, 5, 6$ questions respectively. We need to select $7$ questions such that at least $2$ are sel

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$2700$

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(D) The question paper has three parts $A, B, C$ with $4, 5, 6$ questions respectively. We need to select $7$ questions such that at least $2$ are selected from each part. The possible distributions $(n_A, n_B, n_C)$ are:$1$. $(3, 2, 2)$: $\binom{4}{3} 	imes \binom{5}{2} 	imes \binom{6}{2} = 4 	imes 10 	imes 15 = 600$$2$. $(2, 3, 2)$: $\binom{4}{2} 	imes \binom{5}{3} 	imes \binom{6}{2} = 6 	imes 10 	imes 15 = 900$$3$. $(2, 2, 3)$: $\binom{4}{2} 	imes \binom{5}{2} 	imes \binom{6}{3} = 6 	imes 10 	imes 20 = 1200$Total ways = $600 + 900 + 1200 = 2700$.

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