Subgroup Calculation Strategy
We need to find the number of subgroups containing at least one Indian. A straightforward approach is to calculate the total possible subgroups and subtract the subgroups that contain *no* Indians.
Total Possible Subgroups
The group has 6 people in total (3 Indians + 3 Chinese). The total number of possible subgroups (including the empty set) that can be formed from 6 people is given by $2^n$, where $n$ is the number of people.
Total subgroups = $2^6 = 64$.
Subgroups Without Indians
Subgroups with no Indians must be formed exclusively from the Chinese members. There are 3 Chinese people.
The number of subgroups that can be formed using only the 3 Chinese people is $2^3$.
Subgroups with only Chinese members = $2^3 = 8$.
Calculating Subgroups With At Least One Indian
To find the number of subgroups with at least one Indian, we subtract the number of subgroups containing only Chinese people from the total number of subgroups.
Number of subgroups with at least one Indian = (Total subgroups) - (Subgroups with only Chinese members)
Number of subgroups with at least one Indian = $2^6 - 2^3 = 64 - 8 = 56$.
Therefore, there are 56 subgroups that include at least one Indian.
Important Questions from Permutations and Combinations
Two wizards try to create a spell using all the four elements, water, air, fire, and earth. For this, they decide to mix all these elements in all possible orders. They also decide to work independently. After trying all possible combination of elements, they conclude that the spell does not work.
How many attempts does each wizard make before coming to this conclusion, independently?A police station has six personnel comprising an inspector (I) and five constables ($C_1, C_2, C_3, C_4, C_5$). Normally, a 'raid team' of two, three, or four members is formed depending upon the nature of the raid. As per rule, it is mandatory that the inspector is part of raid team.
Number of distinct raid teams that can be formed is __________________.
(Answer in integer)
Five teams have to compete in a league, with every team playing every other team exactly once, before going to the next round. How many matches will have to be held to complete the league round of matches?
- How many 4-digit positive integers divisible by 3 can be formed using only the digits {1, 3, 4, 6, 7}, such that no digit appears more than once in a number?
- How many five-digit numbers can be formed using the integers 3, 4, 5 and 6 with exactly one digit appearing twice?
