Question

How many ways are there to divide 4 Indian countries and 4 China countries into 4 groups of 2 each such that at least one group must have only Indian countries?

(a) 6

(b) 45

(c) 12

(d) 76

This question was posed to me in class test.

This key question is from Counting in portion Counting of Discrete Mathematics

Answer 1

Right option is (a) 6

Explanation: The number of ways to divide 4+4=8 countries into 4 groups of 2 each is as follows: (^10C2 * ^10C2  * ^10C2 * ^10C2)/4! = 30. Since it is required that at least one group must have only Indian countries, we need to subtract 30 from the number of possible groupings where all 4 groups have 1 Indian country and 1 China country each. This is equivalent to the number of ways to match each of the 4 Indian countries with one China country: 4! = 24. Therefore, the answer is 30 – 24 = 6.