isi-entrance 2011 Q6

isi-entrance · India · solved Combinations & Selection Selection with Group/Category Constraints

Let $A$ be the set $\{ 1,2 , \ldots , 20 \}$. Fix two disjoint subsets $S _ { 1 }$ and $S _ { 2 }$ of $A$, each with exactly three elements. How many 3-element subsets of $A$ are there, which have exactly one element common with $S _ { 1 }$ and at least one element common with $S _ { 2 }$?
(a) 51
(b) 102
(c) 135
(d) 153

(C) Case 1: select one element from $S_1$, one from $S_2$, one from remaining 14: ${}^3C_1 \times {}^3C_1 \times {}^{14}C_1 = 126$ ways. Case 2: select one element from $S_1$ and two from $S_2$: ${}^3C_1 \times {}^3C_2 = 9$ ways. Total $= 126 + 9 = 135$.

Let $A$ be the set $\{ 1,2 , \ldots , 20 \}$. Fix two disjoint subsets $S _ { 1 }$ and $S _ { 2 }$ of $A$, each with exactly three elements. How many 3-element subsets of $A$ are there, which have exactly one element common with $S _ { 1 }$ and at least one element common with $S _ { 2 }$?\\
(a) 51\\
(b) 102\\
(c) 135\\
(d) 153

Paper Questions

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