Understanding Seating Arrangements: A and B Not Together
The problem asks for the number of ways 5 people can be seated in a row such that two specific people, A and B, are not seated next to each other. We can solve this using the principle of complementary counting in permutations.
Step 1: Calculate Total Possible Arrangements
First, find the total number of ways to arrange 5 distinct people in a row without any restrictions. This is calculated using the factorial of the number of people.
Total arrangements = $5!$
$ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 $
There are 120 distinct ways to seat 5 people in a row.
Step 2: Calculate Arrangements Where A and B ARE Together
Next, we calculate the number of arrangements where A and B *are* seated next to each other. We can treat A and B as a single combined unit or block.
Consider the block (AB). Now we have 4 items to arrange: the block (AB) and the other 3 people (C, D, E). The number of ways to arrange these 4 items is $4!$.
$ 4! = 4 \times 3 \times 2 \times 1 = 24 $
Within the block (AB), A and B can switch positions (AB or BA). There are $2!$ ways to arrange A and B within the block.
$ 2! = 2 \times 1 = 2 $
The total number of arrangements where A and B sit together is the product of these two calculations:
Arrangements (A and B together) = (Ways to arrange 4 items) $\times$ (Ways to arrange A and B within the block)
$ \text{Ways (Together)} = 4! \times 2! = 24 \times 2 = 48 $
Step 3: Calculate Arrangements Where A and B ARE NOT Together
To find the number of ways A and B are *not* together, we subtract the number of arrangements where they *are* together from the total number of arrangements.
Arrangements (A and B not together) = Total Arrangements - Arrangements (A and B together)
$ \text{Ways (Not Together)} = 120 - 48 = 72 $
Conclusion
The total number of ways to seat 5 people is 120. The calculation shows that there are 48 ways for A and B to sit together, and 72 ways for A and B *not* to sit together.
The value 48 corresponds to the specific scenario where A and B are adjacent in the seating arrangement.
Similar Questions
- A player holds 15 cards of four suits, of which eight are red and seven are black. The number of diamonds is twice the number of spades, and the number of hearts is equal to the number of diamonds. How many clubs does he hold?
- You have four puzzle books on your shelf, and you decide to arrange them in every possible order, moving one book every 30 seconds. How long will it take to arrange all the books?
- In how many ways can 6 people be seated around a round table?
- If 6 people are seated at a round table, how many distinct seating arrangements are possible?
- In a row of 8 seats, how many ways can 3 people sit next to each other?
- In a row of 8 seats, how many ways can 3 people sit next to each other?
- If 3 men and 2 women are seated in a row, how many seating arrangements are possible such that no woman is seated next to
another woman? - How many ways can 4 people be seated in a circle if two people must sit next to each other?
Important Questions from Permutation and Combination
On a chess board, in how many different ways can 6 consecutive squares be chosen on the diagonals along a straight path ?
There are 6 persons arranged in a row. Another person has to shake hands with 3 of them so that he should not shake hands with two consecutive persons. In how many distinct possible combinations can the handshakes take place ?
In a tournament of Chess having 150 entrants, a player is eliminated whenever he loses a match. It is given that no match results in a tie/draw. How many matches are played in the entire tournament?
The letters A, B, C, D and E are arranged in such a way that there are exactly two letters between A and E. How many such arrangements are possible?
There is a numeric lock which has a 3-digit PIN. The PIN contains digits 1 to 7. There is no repetition of digits. The digits in the PIN from left to right are in decreasing order. Any two digits in the PIN differ by at least 2. How many maximum attempts does one need to find out the PIN with certainty?
