if 11 people are to be accommodated in 11 chairs one in each chair. if 4 of the chairs are too small for 6 of the people, find the number of ways of these people to be seated?
if 11 people are to be accommodated in 11 chairs one in each chair. if 4 of the chairs are too small for 6 of the people, find the number of ways of these people to be seated?
Answer:Let’s break down the problem step by step:
- There are 11 chairs and 11 people.
- 4 of the chairs are too small for 6 of the people. This means there are 7 regular chairs left and 6 people who fit comfortably in them.
Now, let’s calculate the number of ways these 6 people can be seated in the 7 regular chairs:
This is a permutation problem, as the order in which the people sit matters. The formula for permutations is:
[P(n, r) = \frac{n!}{(n-r)!}]
Where (n) is the total number of items (chairs) and (r) is the number of items to be selected (people).
In this case, (n = 7) (number of regular chairs) and (r = 6) (number of people).
P(7, 6) = \frac{7!}{(7-6)!} = \frac{7!}{1!} = 7!
Calculating the factorial:
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
So, there are 5040 ways for these 6 people to be seated in the 7 regular chairs.
Now, let’s consider the 4 chairs that are too small for 6 people. Since there are only 4 small chairs and 6 people who need to sit, it’s not possible to accommodate all of them. Therefore, the number of ways for them to be seated is 0.
Finally, the total number of ways for all the people to be seated is the product of the number of ways for each group of people to be seated:
Total ways = Ways for 6 people in regular chairs × Ways for 6 people in small chairs
Total ways = 5040 × 0 = 0
So, there are 0 ways for all the people to be seated given the constraints.