Five different colored flags will be placed on a pole , one beneath another . The arrangement of the colors indicates the message. How many messages are possible if five flags are to be selected from nine different colored flags?
order matters so:
Evalute the permutation 9P5
A permutation is a way to order or arrange a set or number of things
The formula for a combination of choosing r ways from n possibilities is:
nPr = n!/
(n - r)!
where n is the number of items and r is the number of arrangements.
Plugging in our numbers of n = 9 and r = 5 into the permutation formula:
9P5 = 9!
(9 - 5)!
Remember that n! = n * (n - 1) * (n - 2) * .... * 2 * 1
Calculate the numerator n!:
n! = 9!
9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
9! = 362,880
Calculate the denominator (n - r)!:
(n - r)! = (9 - 5)!
(9 - 5)! = 4!
4! = 4 x 3 x 2 x 1
4! = 24
Calculate our permutation value nPr for n = 9 and r = 5:
9P5 = 362,880/ 24
9P5 = 15,120 answer
Well, there's actually a lot of possible messages, but let's try to paint a picture here. Imagine each color as a punchline to a joke. You've got a pool of nine colors, and you're picking five of them to place on a pole. That's like trying to tell a joke with five punchlines.
Now, imagine how many different jokes you can come up with using those punchlines. It's quite a lot, isn't it? Well, in this case, it's the same concept. Each combination of five flags represents a different message.
So, to calculate the number of possible messages, you can use the combination formula. In this scenario, picking five flags out of nine, the formula is:
9! / (5!(9-5)!)
= 9! / (5!4!)
And when you do the math, you'll find that there are 126 possible messages.
That's a lot of jokes, my friend! I hope you have some good ones lined up for those flags!
To determine how many messages are possible, we need to calculate the number of ways we can select 5 flags out of 9 different colored flags without considering their order.
This can be calculated using the combination formula, which is represented as:
C(n, r) = n! / ((n - r)! * r!)
Where:
n = total number of items to choose from
r = number of items to be chosen
! = factorial
In this case, we have 9 different colored flags and we want to select 5 of them. So, plugging these values into the combination formula gives us:
C(9, 5) = 9! / ((9 - 5)! * 5!)
Calculating the factorials:
9! = 9 * 8 * 7 * 6 * 5!
(9 - 5)! = 4!
Plugging the factorials back into the formula:
C(9, 5) = (9 * 8 * 7 * 6 * 5!) / (4! * 5!)
The 5! terms cancel out:
C(9, 5) = (9 * 8 * 7 * 6) / 4!
Calculating the factorial:
4! = 4 * 3 * 2 * 1
Cancelling again:
C(9, 5) = (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1)
Performing the multiplication:
C(9, 5) = 3024 / 24
Simplifying:
C(9, 5) = 126
Therefore, there are 126 possible messages that can be arranged from selecting 5 flags out of 9 different colored flags.
To determine the number of messages possible, you need to calculate the number of combinations of 5 flags selected from a set of 9 different colored flags. The formula for calculating combinations is given by the binomial coefficient, also known as "n choose k," denoted as C(n, k).
In this case, you want to find C(9, 5). The formula for C(n, k) is:
C(n, k) = n! / (k! * (n-k)!)
Where "n!" represents the factorial of n, which is the product of all positive integers less than or equal to n.
Plugging in the values, we have:
C(9, 5) = 9! / (5! * (9-5)!)
Calculating the factorials:
9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362,880
5! = 5 * 4 * 3 * 2 * 1 = 120
(9-5)! = 4! = 4 * 3 * 2 * 1 = 24
Substituting these values back into the formula:
C(9, 5) = 362,880 / (120 * 24)
Dividing the numerator by the denominator:
C(9, 5) = 3024
Therefore, there are 3024 possible messages that can be formed by selecting 5 flags from 9 different colored flags.