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?
Options:
(A) 56
(B) 52
(C) 48
(D) 44
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2017 Q.64)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (2 Marks)
- **Topic**: Combinatorics & Set Theory (Subsets & Complement Counting)
**2. Detailed Step-by-Step Derivation**:
- We have a group of 6 people comprising:
- 3 Indians ($I_1, I_2, I_3$)
- 3 Chinese ($C_1, C_2, C_3$)
- A "subgroup" (subset) of this group can be formed by choosing any subset of people.
- Total number of non-empty subsets that can be formed from a group of 6 people:
$N_{\text{total non-empty}} = 2^6 - 1 = 64 - 1 = 63$
- We require every chosen subgroup to contain **at least one Indian**:
- The complement condition is choosing a subgroup that contains **no Indians** (i.e., consisting solely of Chinese members).
- Number of non-empty subgroups consisting exclusively of Chinese people chosen from the 3 Chinese:
$N_{\text{no Indians (non-empty)}} = 2^3 - 1 = 8 - 1 = 7$
(These correspond to subgroups of size 1, 2, or 3 formed only from $\{C_1, C_2, C_3\}$).
- Therefore, the number of non-empty subgroups containing at least one Indian is:
$N_{\text{at least one Indian}} = N_{\text{total non-empty}} - N_{\text{no Indians (non-empty)}} = 63 - 7 = 56$
Alternatively, by direct counting:
Number of non-empty subsets of Indians = $2^3 - 1 = 7$.
Any subset of Chinese (including empty) = $2^3 = 8$.
Total subgroups with at least one Indian = $7 \times 8 = 56$.
- **Option-by-Option Evaluation**:
- **Option (A) [CORRECT]**: 56 subgroups contain at least one Indian ($7 \times 8 = 56$).
- **Option (B) [INCORRECT]**: 52 is an arithmetic miscalculation.
- **Option (C) [INCORRECT]**: 48 subtracts all subsets of size 1 and 2.
- **Option (D) [INCORRECT]**: 44 is an erroneous combinatorial estimate.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **A**
- **Scoring Criteria**: +2.0 Marks (Negative marking: -0.67 marks).
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