There are 9 students participating in a spelling bee. In how many ways can the students who go first¸ second¸ and third be chosen?
Well, there are 9 students participating, so we could have nine different acts of picking someone from the group and giving them a giant foam hand that says "Go First!" on it. And then we could have eight different acts of picking someone else and giving them a foam hand that says "Go Second!" on it. And finally, we could have seven different acts of picking a third person and giving them a foam hand that says "Go Third!" on it.
So, using the fundamental principle of counting, we multiply these possibilities together: 9 x 8 x 7 = 504 different ways to choose the students who go first, second, and third in the spelling bee. That's a lot of foam hands!
To determine the number of ways the students who go first, second, and third can be chosen, we can use the concept of permutations.
Since the order matters (being first, second, or third), we can use the formula for permutations:
nPr = n! / ((n-r)!)
Where n represents the total number of students (9) and r represents the number of students to be chosen (3).
Plugging in the values, we get:
9P3 = 9! / (9-3)!
= 9! / 6!
Calculating further:
9! = 9 * 8 * 7 * 6!
6! = 6 * 5 * 4 * 3 * 2 * 1
Substituting the above values back into the formula:
9P3 = (9 * 8 * 7 * 6!) / (6 * 5 * 4 * 3 * 2 * 1)
= (9 * 8 * 7) / (3 * 2 * 1)
= 504
Therefore, there are 504 ways the students who go first, second, and third can be chosen.
To determine the number of ways to choose the students who go first, second, and third in the spelling bee, we can use the concept of permutations.
Permutations represent the number of ways to arrange or select objects in a specific order.
In this case, we have 9 students participating, and we need to choose three of them to go in a specific order (first, second, and third).
To find the number of permutations, we can use the formula for permutations of n objects taken r at a time:
P(n, r) = n! / (n - r)!
where n is the total number of objects, r is the number of objects we want to choose, and the exclamation mark (!) represents the factorial operation.
In this case, n = 9 (total number of students) and r = 3 (number of students to select). Plugging these values into the formula:
P(9, 3) = 9! / (9 - 3)!
= 9! / 6!
We can simplify this further:
9! = 9 x 8 x 7 x 6!
6! = 6 x 5 x 4 x 3 x 2 x 1
Substituting these values back into the formula:
P(9, 3) = (9 x 8 x 7 x 6!) / (6 x 5 x 4 x 3 x 2 x 1)
Calculating further:
P(9, 3) = (9 x 8 x 7) / (6 x 5 x 4)
= 504
Therefore, there are 504 ways to choose the students who go first, second, and third in the spelling bee.
first one 9 choices
second one 8 choices
third one 7 choices
9 * 8 * 7
or look up permutations
number of ways to arrange n things taken r at a time
P(n,r) = n! / (n-r)!
here
P(9,3) = 9! /(9-3)! = 9!/6! = 9*8*7 *6!/6! = 9*8*7 again