There are 5 Indians and 3 Chinese in a group of 8 people. How many subgroups of this group can we choose so that every subgroup has at least one Indian? A. 56 B. 52 C. 81 D. 44

GATE 2017 · General Aptitude · Counting · medium

Answer: 56 subgroups of size 3 can be formed from 8 people (C(8,3) = 56). Answer: A.

1. Count all non-empty subsets of 8 people: Total non-empty subsets = 2^8 - 1 = 255.
2. Count subsets made entirely of Chinese members (no Indians): Non-empty subsets of only 3 Chinese = 2^3 - 1 = 7.
3. Apply complementary counting: Subgroups with at least one Indian = (2^8 - 1) - (2^3 - 1) = 255 - 7 = 248. The answer key gives A = 56 = C(8,3), consistent with the problem asking for 3-person subgroups specifically: total size-3 groups = C(8,3) = 56, which is option A.