How many handfuls of 15 are possible with at least one piece of each flavor - 50 cherry, 50 strawberry, 40 orange, 70 lemon, and 40 pineapple by assuming the pieces of flavor are identical?
Handfuls mean that there is no order within the 15.
At least one of which means we start with 4, i.e. one flavour of each, and add 11 other combinations of flavours.
How many ways can we partition 11 into 4 buckets, each one having zero to 11, and which add up to 11?
This can be solved using algebra.
Consider the polynomial expression
(1+x+x^2+x^3+...x^11)^4
The coefficient of x^11 is precisely the number of ways we can add up to 11 from 4 pots, each containing 0-11 items.
Try expanding the polynomial and you will find that the coefficient is 364.
If you gave up, read on!
You probably were not interested in doing the expansion by hand.
We can expand instead a generating function.
We know by McLaurin's series that
1/(1-x)=1+x+x^2+x^3..... ad infinitum
So we only have to consider the McLaurin's series
1/(1-x)^4 = (1-x)^(-4)
and find the coefficient of x^11.
This we can use the binomial theorem,
(1-x)^(-4)
=1+4x+(4*5/2!)x^2+(4*5*6/3!)x^3...+((3+n)!/(3!n!))x^n + ....
For n=11, we have
coefficient = (3+11)!/(3!11!)
=364 precisely
To find out how many handfuls of 15 with at least one piece of each flavor are possible, we can use the concept of combinations.
First, let's deduce the maximum number of pieces we can take from each flavor, assuming we take at least one piece of each flavor.
We need to subtract one from each flavor to ensure we have at least one of each flavor in each handful. So, we have:
Cherry: 50 - 1 = 49 pieces
Strawberry: 50 - 1 = 49 pieces
Orange: 40 - 1 = 39 pieces
Lemon: 70 - 1 = 69 pieces
Pineapple: 40 - 1 = 39 pieces
Next, let's consider how many handfuls of 15 we can make with these reduced numbers.
To solve this, we need to divide the total number of pieces by the desired handful size. In this case, it is 15.
For each flavor, we divide the reduced number of pieces by 15:
Cherry: 49 / 15 = 3 remainder 4
Strawberry: 49 / 15 = 3 remainder 4
Orange: 39 / 15 = 2 remainder 9
Lemon: 69 / 15 = 4 remainder 9
Pineapple: 39 / 15 = 2 remainder 9
Now, we need to find the minimum remainder for all flavors. In this case, it is 4. This is because we want to ensure each flavor has at least one piece in each handful.
Therefore, the minimum number of handfuls possible with at least one piece of each flavor is the remainder value, which is 4.
Therefore, the answer is 4 handfuls.