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
- 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?
