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(A) Let the three sections be $S_1, S_2, S_3$,each with $4$ questions. The candidate must select $5$ questions such that at least one is selected from

Vedclass · Accepted Answer

$624$

Vedclass · Answer

(A) Let the three sections be $S_1, S_2, S_3$,each with $4$ questions. The candidate must select $5$ questions such that at least one is selected from each section. The possible distributions of questions across the three sections are $(1, 1, 3)$ or $(1, 2, 2)$ in any order.Case $1$: Distribution $(1, 1, 3)$ in any order.The number of ways to choose the sections is $\frac{3!}{2!} = 3$.The number of ways to select the questions is $^4C_1 	imes ^4C_1 	imes ^4C_3 = 4 	imes 4 	imes 4 = 64$.Total ways for this case $= 3 	imes 64 = 192$.Case $2$: Distribution $(1, 2, 2)$ in any order.The number of ways to choose the sections is $\frac{3!}{2!} = 3$.The number of ways to select the questions is $^4C_1 	imes ^4C_2 	imes ^4C_2 = 4 	imes 6 	imes 6 = 144$.Total ways for this case $= 3 	imes 144 = 432$.Total number of ways $= 192 + 432 = 624$.

There are three sections in a question paper,each section containing $4$ questions. If a candidate has to answer exactly $5$ questions from this paper such that at least one question is answered from each section,then the number of ways in which a candidate can make the choice of questions is

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