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
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 five-digit numbers can be formed using the integers 3, 4, 5 and 6 with exactly one digit appearing twice?
- 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)
- There are 3 Indians and 3 Chinese in a group of 6 people. How many subgroups of this group can we choose so that every subgroup has at least one Indian?
- The number of elements of order 3 in the symmetric group $S_6$ is________
