METHOD 1: Simple Complement Method
Total Subgroup = 2^6
Subgroup of size 0 = 2^3 =8
So subgroup of this group can we choose so that every subgroup has at least one Indian = 64 - 8 = 56
METHOD 2 : Combinatorics Method :
A subgroup can have 1, 2, or 3 Indians, along with any number (1, 2, or 3) of Chinese people.
Ways to choose Chinese participants:
$\text{Chinese Choices} = \binom{3}{0} + \binom{3}{1} + \binom{3}{2} + \binom{3}{3} = 1 + 3 + 3 + 1 = 8$
Case 1: Subgroup with exactly 1 Indian
$\text{Ways} = \binom{3}{1} \times \text{Chinese Choices} = 3 \times 8 = 24$
Case 2: Subgroup with exactly 2 Indians
$\text{Ways} = \binom{3}{2} \times \text{Chinese Choices} = 3 \times 8 = 24$
Case 3: Subgroup with exactly 3 Indians
$\text{Ways} = \binom{3}{3} \times \text{Chinese Choices} = 1 \times 8 = 8$
Total Valid Subgroups:
$\text{Total} = 24 + 24 + 8 = 56$
