An ice cream stand has nine different flavors. A group of children come to the stand and each buys a double scoop cone with two flavors of ice cream.If none of the children choose the same combination of flavors ,but all possible combinations of flavors are used, how many children are there?
Flavors: Vanilla, Maple, Chocolate,Tiger, Raspberry,Strawberry, Coffee, Moon Mist, Cherry Vanilla
Well, if there are nine different flavors and each child chooses a combination of two flavors, we can think of it as each child picking two flavors from a total of nine.
Since order doesn't matter for the combinations, we can use the combination formula, which is nCr = n! / r!(n-r)!. In this case, n = 9 (flavors) and r = 2 (combination of two flavors).
So, using the formula, we get 9C2 = 9! / 2!(9-2)! = 9! / 2!7! = (9 * 8) / 2 = 72 / 2 = 36.
Therefore, there are 36 children in total.
To find out the number of children, we need to determine the number of unique combinations of two flavors that can be made from the nine available flavors.
The number of combinations of "n" items taken "r" at a time can be calculated using the formula:
C(n, r) = n! / (r! * (n - r)!)
Where "!" denotes the factorial operation.
In this case, we have 9 flavors and we want to find the number of unique combinations of 2 flavors out of those 9.
C(9, 2) = 9! / (2! * (9 - 2)!)
= 9! / (2! * 7!)
= (9 * 8 * 7!) / (2! * 7!)
= (9 * 8) / 2!
= 72 / 2
= 36
Therefore, there are 36 children in total.
To determine the number of children, we need to find the total number of unique combinations of two flavors that can be made from the nine available flavors.
The formula to find the number of combinations of items without repetition is:
C(n, r) = n! / (r!(n-r)!),
where:
- C(n, r) represents the number of combinations of n items taken r at a time,
- n! represents the factorial of n (the product of all positive integers from n to 1),
- r! represents the factorial of r, and
- (n-r)! represents the factorial of (n-r).
In this case, we have nine flavors available (n = 9) and each child buys a double scoop cone with two flavors (r = 2).
Using the formula, we can calculate the number of children:
C(9, 2) = 9! / (2!(9-2)!)
= 9! / (2!7!)
= (9 * 8 * 7!)/(2! * 7!)
= (9 * 8)/(2 * 1)
= 72 / 2
= 36
Therefore, there are 36 children in the group.
There are 9 things to choose from and we choose 2 of them
There are C(9,2) such choices or 36 choices,
which must match the number of children