How many different ways can 5 people be seated on a bench?
Assuming there are 5 places on the bench, and you place the people one seat at a time.
Number of people available for first seat = 5
Number of people available for second seat = 4
....
Number of people available for the last seat = 1
Total number of arrangements = 5! = 5.4.3.2.1
120
5*4*3*2*1
Ah, the classic question of seating arrangements. Well, let's see. If I remember correctly, there are actually quite a few ways to arrange those five people on a bench. In fact, there are 120 different ways! That's right, it's like a game of musical chairs on steroids. Just make sure none of them try to sit on each other's laps – that could get a little messy. So, go ahead and let the seating shenanigans begin!
To calculate the number of different ways 5 people can be seated on a bench, we need to use the concept of permutations.
A permutation is an arrangement or order of a set of objects. In this case, the objects are the 5 people. The formula to calculate the number of permutations is:
P(n, r) = n! / (n - r)!
Where:
- P(n, r) represents the number of permutations of n objects taken r at a time.
- n! represents the factorial of n, which is the product of all positive integers less than or equal to n.
In this case, we have 5 people to be seated on a bench, which means we need to find the number of permutations taken all at a time (r = 5). Plugging the values into the formula:
P(5, 5) = 5! / (5 - 5)!
Since 5 - 5 equals 0, anything divided by 0 is undefined. Therefore, we need to redefine the formula for P(n, r) when r = 0:
P(n, 0) = 1
Using this formula, we get:
P(5, 5) = 5! / 0! = 5!
Calculating the factorial of 5:
5! = 5 x 4 x 3 x 2 x 1 = 120
Therefore, there are 120 different ways 5 people can be seated on a bench.