a The number of possible performance orders of all 7 gymnasts is the number of their permutations.
B
b The number of possible performance orders of all 7 gymnasts is the number of their permutations.
A
a 1/35
B
b 1/210
Practice makes perfect
a We are given a table with the finalists of the floor exercises competition
Floor Exercises Finalists
Eliza Hernandez
Kimi Kanazawa
Cecilia Long
Annie Montgomery
Shenice Malone
Caroline Smith
Jessica Watson
We are asked to find the probability that Cecilia, Annie, and Kimi are the first 3 gymnasts to perform, in any order. To do that we will compare the number of favorable outcomes with the number of possible outcomes.
P=Favorable outcomes/Possible outcomesThe number of possible outcomes is the number of all possible orders in which the gymnasts can perform.
The number of possible orders is equal to the number of permutations of the 7 gymnasts, which is 7!. Therefore, the number of possible outcomes is 7!.
Possible outcomes= 7!
The number of favorable outcomes is the number of possible orders of performances in which the first 3 gymnasts are Cecilia, Annie, and Kimi. The order of Cecilia, Annie, and Kimi does not matter.
The last 4 gymnasts can be ordered in 4! different ways. Also, there are 3! possibilities of ordering Cecilia, Annie, and Kimi.
Therefore, using the Fundamental Counting Principle we can calculate the number of favorable outcomes by multiplying 3! by 4!.
Favorable outcomes= 3!*4!
Now, we are ready to calculate the probability.
P=Favorable outcomes/Possible outcomes=3!*4!/7!
Let's simplify this quotient.
The probability that the first 3 gymnasts to perform will be Cecilia, Annie, and Kimi, in any order, is 135.
b This time we want to calculate the probability that Cecilia will perform first, Annie will be second, and Kimi will be third. To do that, we will once again compare the number of favorable outcomes with the possible outcomes.
P=Favorable outcomes/Possible outcomes
The number of possible outcomes is the number of possible orders of performance of all the 7 gymnasts. We already found that number in Part A: it is equal to 7!.
Possible outcomes= 7!The favorable outcome is when Cecilia performs first, Annie is second, and Kimi is third.
The order of performances of the first 3 gymnasts is set. The last four gymnasts can be arranged in 4! different ways. Therefore, the number of favorable outcomes is also 4!.
Favorable outcomes= 4!
Now we can calculate the probability.
P=Favorable outcomes/Possible outcomes=4!/7!
Let's simplify this.
