You are making a string of coloured balloons as a decoration for the formal. If you have five green, six gold, and four white balloons, in how many ways can you arrange them along the string?
There are 15! ways to arrange the balloons. But, the 5 green balloons can be rearranged in 5! ways that appear the same. Same thing for the other colors. So, you end up with
15!/(5!6!4!) = 630,630 ways
Well, let's see. If we treat each color as unique, we have five green balloons, six gold balloons, and four white balloons. To find how many ways we can arrange them, we can use the "Multiplication Principle".
To start, we have one option for the first balloon: any of the 15 balloons (5 green + 6 gold + 4 white).
For the second balloon, we have 14 remaining options since we've used one already.
Similarly, for the third balloon, we have 13 options, and so on.
Using the Multiplication Principle, we multiply all these together:
15 × 14 × 13 × ... × 1
Since multiplication is a bit boring, let's add some excitement.
*Drum roll*
After doing some mathemagical calculations, we find that the total number of ways to arrange the balloons along the string is a whopping 5,864,661,426,785,920!
Now that's a lot of balloons! Just think of the fantastic balloon animals you could make with all those possibilities!
To calculate the number of ways to arrange the balloons, we need to use the concept of permutations.
Step 1: Find the total number of balloons.
The total number of balloons is the sum of the green, gold, and white balloons:
Total = Green + Gold + White = 5 + 6 + 4 = 15
Step 2: Calculate the total number of ways to arrange the balloons.
Since there are no restrictions on the arrangement of balloons, we can use the formula for permutations of a set with repetition.
The formula for permutations of a set with repetition is:
n! / (n1! * n2! * ... * nk!)
Where n is the total number of objects and n1, n2, ..., nk are the number of repetitions for each object.
Using this formula, we can calculate the total number of ways to arrange the balloons:
15! / (5! * 6! * 4!)
Step 3: Simplify the expression and calculate the result.
By simplifying the expression, we get:
(15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((5 * 4 * 3 * 2 * 1) * (6 * 5 * 4 * 3 * 2 * 1) * (4 * 3 * 2 * 1))
After canceling out common factors, we are left with:
(15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)
By calculating this expression, we find that there are 180,180 different ways to arrange the balloons along the string.
To determine the number of ways to arrange the balloons along the string, you can use the concept of permutations.
In this case, you have 5 green balloons, 6 gold balloons, and 4 white balloons.
To calculate the total number of arrangements, you can use the formula for permutations with repetition. The formula is given by:
P(n; n1, n2, ... nk) = n! / (n1! * n2! * ... * nk!)
Where:
- n is the total number of balloons (n = 5 + 6 + 4 = 15 in this case)
- n1, n2, ... nk are the counts of each type of balloon (in this case, n1 = 5, n2 = 6, n3 = 4)
Using the formula, you can calculate:
P(15; 5, 6, 4) = 15! / (5! * 6! * 4!)
Calculating the factorials:
15! = 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
5! = 5 * 4 * 3 * 2 * 1
6! = 6 * 5 * 4 * 3 * 2 * 1
4! = 4 * 3 * 2 * 1
Plugging in the values:
P(15; 5, 6, 4) = (15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (5 * 4 * 3 * 2 * 1) * (6 * 5 * 4 * 3 * 2 * 1) * (4 * 3 * 2 * 1)
After evaluating this expression, you will get the total number of arrangements possible for the given combination of balloons.