Four persons P, Q, R and S are to be seated in a row. R should not be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:
Calculating Seating Arrangements with Constraints
We need to find the number of distinct ways to seat 4 persons (P, Q, R, S) in a row, with the constraint that R cannot be in the second position from the left.
Total Possible Arrangements
Without any restrictions, the 4 persons can be arranged in the 4 seats in $4!$ ways.
Calculation: $4! = 4 \times 3 \times 2 \times 1 = 24$. So, there are 24 total possible seating arrangements.
Arrangements Violating the Constraint
Now, let's calculate the number of arrangements where R is seated in the second position (the restricted position).
If R is fixed in the second position, the remaining 3 persons (P, Q, S) can be arranged in the remaining 3 positions (1st, 3rd, 4th) in $3!$ ways.
Calculation: $3! = 3 \times 2 \times 1 = 6$. There are 6 arrangements where R is in the second position.
Distinct Arrangements Satisfying the Constraint
To find the number of arrangements where R is *not* in the second position, we subtract the number of violating arrangements from the total number of arrangements.
Possible Arrangements = Total Arrangements - Arrangements where R is in the second position
Calculation: $24 - 6 = 18$.
Therefore, there are 18 distinct seating arrangements possible where R is not seated at the second position from the left.
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?
