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)
Raid Team Formation Calculation
The problem requires calculating the number of distinct raid teams possible under specific conditions.
- Total personnel: 1 Inspector (I) + 5 Constables ($C_1, C_2, C_3, C_4, C_5$).
- Raid team size can be 2, 3, or 4 members.
- Constraint: The Inspector (I) must always be included in the team.
Constable Selection Logic
Since the Inspector is mandatory, the remaining team members must be chosen from the 5 constables. The number of constables to select depends on the required team size.
Calculating Teams for Each Size
We calculate the possibilities for each allowed team size:
- Team Size 2: The team consists of the Inspector and 1 constable. The number of ways to choose 1 constable from 5 is given by the combination formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. Number of ways = $\binom{5}{1} = \frac{5!}{1!(5-1)!} = 5$.
- Team Size 3: The team consists of the Inspector and 2 constables. Number of ways = $\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10$.
- Team Size 4: The team consists of the Inspector and 3 constables. Number of ways = $\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10$.
Total Distinct Raid Teams
The total number of distinct raid teams is the sum of the possibilities for each team size.
Total Teams = (Ways for Size 2) + (Ways for Size 3) + (Ways for Size 4)
Total Teams = $5 + 10 + 10 = 25$.
Important Questions from Permutations and Combinations
- 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?
Three distinct sets of indistinguishable twins are to be seated at a circular table that has 8 identical chairs. Unique seating arrangements are defined by the relative positions of the people.
How many unique seating arrangements are possible such that each person is sitting next to their twin?
- Mixed species flocks of birds include social and solitary species. There are 5 social species and 10 solitary species in a forest. Flocks always have a total of 5 species, of which 2 are social and 3 are solitary. The number of types of flocks with unique species composition is ________.(Answer in integer)
- How many five-digit numbers can be formed using the integers 3, 4, 5 and 6 with exactly one digit appearing twice?
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?
