a If n objects are arranged relative a fixed reference point then the arrangements can be treated as linear, making the number of permutations n!.
B
b Compare the number of favorable outcomes to the number of possible outcomes. The number of possible outcomes is the answer to Part A.
C
c Use the formula for permutations to find the number of possible arrangements of 6 people under the age of 8 in the 9 seats of the carousel.
A
a 362 880
B
b 1/36
C
c 2/3
Practice makes perfect
a We are asked to find the number of ways in which 9 people can fill the seats of a carousel. In total, the carousel has 9 seats — 7 horses and one bench seat that will hold two people.
We can treat the bench seat as a fixed reference point of the carousel.
Let's recall the formula for the number arrangements of objects in a circle with a fixed reference point.
If n objects are arranged relative to a fixed reference point, then the arrangements can be treated as linear, making the number of permutations n!.
In our case there are 9 people on the carousel, so there are 9! possible permutations. Let's calculate the value of 9!.
We found that the number of ways in which 9 people can fill the seats of a carousel is 362 880.
b We are asked to find the probability that you and your friend will end up on the bench seat if the seats of the carousel are filled randomly. To do that, we will compare the number of favorable outcomes to the number of possible outcomes.
P=Favorable outcomes/Possible outcomes
The number of possible outcomes is the the total number of ways in which 9 people can fill the seats of the carousel. We already found that number in Part A — it is equal to 362 880.
Possible outcomes= 362 880
Now let's find the number of favorable outcomes, which is the number of possible situations in which you and your friend end up on the bench seat.
Notice that there are 2 ways you and your friend can seat on the bench seat.
The seats on the horses can be filled by the other 7 people in 7! different ways.
Therefore, in total there are 7! + 7!=2*7! different situations in which you and your friend end up in the bench seat. This means that the number of favorable outcomes is 2*7!.
Favorable outcomes= 2*7!
Now we are ready to calculate the probability that you and your friend will be together on the bench seats.
P=Favorable Outcomes/Possible Outcomes=2*7!/362 880
Let's simplify this quotient!
c We are told that 6 people that will go on a carousel are under the age of 8. On the carousel, one of the horses is broken and it does not go up or down.
We are asked to find the probability that a person under the age of 8 (a kid) will end up on the broken horse. To do that, we will again compare the number of favorable outcomes to the number of possible outcomes.
P=Favorable Outcomes/Possible Outcomes
The number of possible outcomes is the number of all the ways of arranging 6 kids on the 9 seats of the carousel. We can find that number using permutations.
Permutations
The number of permutations of n distinct objects taken r at a time is denoted by _nP_r and can be calculated using the following formula.
_nP_r=n!/( n- r)!
There are 9 possible seats and we want to assign 6 of them to the kids. Therefore, let's substitute n= 9 and r= 6 into the formula.
The number of possible outcomes is 60 480.
Possible Outcomes= 60 480
The favorable outcome is when one of the 6 kids sits on the broken horse.
Since there are 6 different kids, the number of possibilities in which a kid sits on the broken horse is 6. The other 5 kids can fill the other 8 seats in _8P_5 different ways. Therefore, according to the Fundamental Counting Principle the number of favorable outcomes is 6* _8P_5. Let's use the formula for permutations again.
Favorable Outcomes= 6* _8P_5 ⇕ Favorable Outcomes= 6*8!/( 8- 5)!
We can simplify this expression.
We found that the number of favorable outcomes is 40 320. We are finally ready to calculate the probability that one of the kids will end up on the broken horse.
P=Favorable outcomes/Possible outcomes=40 320/60 480
We can simplify this quotient.
