I have five apples and ten oranges. If a fruit basket must contain at least one piece of fruit, how many kinds of fruit baskets can I make? (The apples are identical and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and it doesn't matter how the fruit are arranged in the basket.)
Are the fruit baskets identical too?
Not that it makes much difference. You would just have to double the answer if they are not.
I'm just going to look at one fruit basket, it can have
5apples and 0oranges
5apples and 1oranges
5apples and 2oranges
5apples and 3oranges
5apples and 4oranges
5apples and 5oranges
5apples and 6oranges
5apples and 7oranges
5apples and 8oranges
5apples and 9oranges
That is 9 ways it can have 5 apples
There are 10 ways it can have 4 apples
There are 10 ways it can have 3 apples
There are 10 ways it can have 2 apples
There are 10 ways it can have 1 apples
There are 9 ways it can have 0 apples
So that is
If the baskets are different then you will have to double this answer.
By analyzing this question, it’s easy to see the distinguishable fruit is irrelevant here. An orange is the same as an apple. The selection of any fruit creates a fruit basket.
The solution to this is the number of partitions of 15.
Calculating the number of partitions (n) requires an understanding of Generating Functions—specifically Euler’s function
For small numbers (n) it’s easy, but sometimes laborious, to do them by hand.
WolfRam Alpha will solve these and list the partitions.
For n = 15, partitions = 176 <----- Number of ways to divide five apples and ten oranges into one or more baskets.
15+0 | One basket with 15 fruits
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 | Fifteen baskets, each with one fruit
14+1 | Two baskets, one with 14, and one with one fruit(s)
13+2 |Two baskets, one with 13, and one with two fruits.
13+1+1 |Three baskets, one with 13, and 2 with one fruit each
12+3 | etc
12+2+1 | etc
12+1+1+1 | etc
11+4 | etc
11+3+1 | etc
11+2+2 | etc
Ginger I expect you have answered some other version of this question. It is not clearly written.
But even though that is true I admit to haveing read the quesion incorrectly.
I thought the question said 2 fruit baskets. Since the question does not appeared to have been edited i must conclude that I read it wrongly in the first place.
My original answer assumed that there are only 2 fruit baskets.
Now I will interpret it as not mattering how many fruit baskets there are, i am just fining how many individual combinations of fruite it is possible for any one fruit basket to have.
I have five apples and ten oranges. If a fruit basket must contain at least one piece of fruit, how many kinds of fruit baskets can I make? (The apples are identical and the oranges are identical.)
What can I have in a individual fruit basket..
0 apples and 1-10 oranges that is 10 posibilities
1 apple and 0-10 oranges that is 11 possibilities
2 apple and 0-10 oranges that is 11 possibilities
3 apple and 0-10 oranges that is 11 possibilities
4 apple and 0-10 oranges that is 11 possibilities
5 apple and 0-10 oranges that is 11 possibilities
Total number of what I can put in a basket is 65 possibilities.
It is a little higher than I got last time becasuse I have not considered what may or may not be in any other basket. There may not be any other fruit basket.
Ginger I expect you have answered some other version of this question.
Yes, I’ve answered many questions related to statistics and combinatorics. I’ve just completed my Masters in Psychology (M.S.) (and Art History (M.A.)). My principle focus is on Abnormal Behavior and Analysis, and this requires a thorough understanding of statistical methods, including combinatorics relating to selection methods. This is why I do well in answering statistical/combinatorics related questions.
My beginning in anything relating to upper-level mathematics was coming to this forum, and particularly meeting Lancelot Link. Before that, I knew only the (rote) probabilities relating to drawing specific poker-hands; I didn’t know how to calculate them mathematically.