Question

There are 3 Indians and 3 Chinese in a group of 6 people. How many subgroups of this group can we choose so that every subgroup has at least one Indian ?

• A. $56$
• B. $52$
• C. $48$
• D. $44$

Answer 1

The correct option is A $56$

Subgroups containing only Indians
${=}^3{C}_1{+}^3{C}_2{+}^3{C}_3=3+3+1=7$

Subgroups containing one Indian and rest Chinese
${=}^3{C}_1\left[{}^3{C}_1{+}^3{C}_2{+}^3{C}_3\right]$
$=3[3+3+1]=21$

Subgroups containing two Indian and remaining Chinese
${=}^3{C}_2\left[{}^3{C}_1{+}^3{C}_2{+}^3{C}_3\right]=21$

Subgroups containing three Indian and remaining Chinese
${=}^3{C}_3\left[{}^3{C}_1{+}^3{C}_2{+}^3{C}_3\right]=7$

Total number of subgroups
$=7+21+21+7=56.$