The problem asks us to determine the number of possible subgroups from a group of 6 people, consisting of 3 Indians and 3 Chinese, such that every chosen subgroup includes at least one Indian. This is a classic combinatorial problem that can be solved using the principle of inclusion-exclusion or, more simply, by subtracting unwanted cases from the total possibilities.

Subgroups Problem Overview

We are given a group of 6 people, specifically divided into two distinct categories:

• 3 Indians (let's denote them as \(I_1, I_2, I_3\))
• 3 Chinese (let's denote them as \(C_1, C_2, C_3\))

Our goal is to form subgroups. A subgroup can be of any size, from just one person up to all six people. The critical condition is that each selected subgroup must contain "at least one Indian."

Problem Analysis: Group Composition

To ensure a subgroup has at least one Indian, it means we cannot have subgroups that consist solely of Chinese people. This suggests a straightforward approach: calculate the total number of possible subgroups and then subtract the number of subgroups that violate our condition (i.e., those with no Indians).

Approach for Subgroup Formation

The number of subgroups that can be formed from a set of 'n' distinct items is \(2^n\). This is because for each item, there are two choices: either include it in the subgroup or exclude it. This method naturally accounts for subgroups of all possible sizes, including the empty set (which usually isn't considered a "subgroup of people" in these contexts, but the formula \(2^n\) includes it. However, in this problem, removing the "no Indian" case implicitly handles the empty set as well, since it contains no Indians).

Detailed Calculation Steps

We will use the complementary counting principle:

Number of subgroups with at least one Indian = (Total number of possible subgroups from 6 people) - (Number of subgroups with no Indians)

1. Calculate Total Possible Subgroups:

   The total number of people in the group is 6 (3 Indians + 3 Chinese). The total number of subgroups that can be formed from these 6 people is given by \(2^n\), where \(n=6\).

   Total Subgroups = \(2^6\)

   Total Subgroups = \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\)

2. Calculate Subgroups with No Indians:

   A subgroup with no Indians means that all members of the subgroup must be Chinese. We have 3 Chinese people. The number of subgroups that can be formed using only these 3 Chinese people is \(2^k\), where \(k=3\).

   Subgroups with No Indians = \(2^3\)

   Subgroups with No Indians = \(2 \times 2 \times 2 = 8\)

3. Subtract to Find Required Subgroups:

   Now, subtract the subgroups with no Indians from the total number of subgroups.

   Subgroups with at least one Indian = Total Subgroups - Subgroups with No Indians

   Subgroups with at least one Indian = \(64 - 8\)

   Subgroups with at least one Indian = \(56\)

Illustrative Subgroup Counts

Here's a summary of the calculations:

Final Subgroup Count

Based on the calculations, there are 56 subgroups that can be chosen from the group of 6 people such that every subgroup has at least one Indian.