If 9 red balloons and 6 blue balloons are to be distributed to 4 children, how many distributions are possible if every child must receive a balloon of each color?
12
To find the number of distributions, we need to determine how many ways we can distribute the red balloons to the children, and then multiply that by the number of ways we can distribute the blue balloons.
For the red balloons:
We have 9 red balloons to distribute among 4 children, and each child must receive at least one red balloon. This is equivalent to distributing 9-4=5 additional red balloons among the 4 children.
To find the number of ways to distribute the additional red balloons, we can use the stars and bars method. In this case, we have 5 additional red balloons and 4 children, so we need to place 5 "bars" between the balloons to divide them among the children.
The number of distributions can be calculated using the formula (n+k-1) choose (k-1), where n is the number of additional red balloons and k is the number of children. In this case, we have (5+4-1) choose (4-1) = 8 choose 3 = 8! / (3! * 5!) = 56.
For the blue balloons:
We have 6 blue balloons to distribute among the 4 children, and each child must receive at least one blue balloon. This is equivalent to distributing 6-4=2 additional blue balloons among the 4 children.
Using the stars and bars method again, we have 2 additional blue balloons and 4 children, so we need to place 2 "bars" between the balloons to divide them among the children.
The number of distributions can be calculated using the formula (n+k-1) choose (k-1), where n is the number of additional blue balloons and k is the number of children. In this case, we have (2+4-1) choose (4-1) = 5 choose 3 = 5! / (2! * 3!) = 10.
To find the total number of distributions, we need to multiply the number of distributions for the red balloons by the number of distributions for the blue balloons:
Total number of distributions = 56 * 10 = 560.
Therefore, there are 560 possible distributions if every child must receive a balloon of each color.
To solve this problem, we can use the concept of permutations.
First, let's consider the red balloons. We have 9 red balloons to distribute, and every child must receive a red balloon. Since there are 4 children, we can think of this as distributing 9 red balloons among 4 children, which means we need to find the number of permutations of 9 objects taken 4 at a time.
The formula for permutations is given by nPr = n! / (n - r)!, where n is the total number of objects and r is the number of objects taken at a time.
In this case, we have 9 red balloons and we want to distribute them among 4 children. Therefore, n = 9 and r = 4. Plugging these values into the formula, we get:
9P4 = 9! / (9 - 4)! = 9! / 5!
Simplifying further:
9P4 = (9 × 8 × 7 × 6 × 5!) / 5!
The 5! cancels out, leaving us with:
9P4 = 9 × 8 × 7 × 6 = 3,024
So, there are 3,024 possible distributions of the red balloons among the 4 children.
Now, let's consider the blue balloons. We have 6 blue balloons to distribute, and every child must receive a blue balloon. Using the same logic as before, we need to find the number of permutations of 6 objects taken 4 at a time.
6P4 = 6! / (6 - 4)! = 6! / 2!
Simplifying further:
6P4 = (6 × 5 × 4 × 3 × 2!) / 2!
The 2! cancels out, leaving us with:
6P4 = 6 × 5 × 4 × 3 = 360
So, there are 360 possible distributions of the blue balloons among the 4 children.
To find the total number of distributions, we need to multiply the number of possible distributions of the red balloons by the number of possible distributions of the blue balloons.
Total number of distributions = 3,024 × 360 = 1,089,280
Therefore, there are 1,089,280 possible distributions of the balloons if every child must receive a balloon of each color.