Combination/Permutation Problem

  • Thread starter Seda
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In summary, you can make a basket containing 7 fruit from apples, oranges, and bananas. If you choose to put in a basket, 0, 1, 2, 3, 4, 5, 6, or 7 apples, the number of bananas is fixed. If you choose to put 1 apple in, the number of oranges is now fixed. If you choose to put 2 apples in, the number of oranges and bananas are now fixed.
  • #1
Seda
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Homework Statement



What is the number of distinguishable fruit baskets, each with seven fruit, can you make with apples, oranges, and bananas.

Homework Equations



Knowledge of Combinations/Permutations

The Attempt at a Solution



Well, to me, this is weird because this is essentially a problem where there is replacement. And obviously, I need to look at combinations, not permutations, because I need distinct baskets.

However, I know the general formula for combination when it's read "n choose k"
, but the way this question is worded, it seems k is bigger than n.

Can I just get some help on how to think about this in terms of combination with replacement or something? Help is appreciated.
 
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  • #2
Okay, I've learned that this is a combination with repetition, which the formula is C (n+k-1; k)



but can anyone here tell me how that formula is derived? I've look at it for 30 minutes and I am clueless.
 
  • #3
Uh, don't double post, ok?
 
  • #4
:uhh:
 
Last edited:
  • #5
You are to make a basket containing 7 fruit from apples, oranges, and bananas.

You can choose to put in a basket, 0, 1, 2, 3, 4, 5, 6, or 7 apples: 8 choices.

If you chose to put 0 apples in you could choose
0, 1, 2, 3, 4, 5, 6, or 7 oranges. For each of those, the number of bananas is now fixed.

If you chose to put 1 apple in you could choose
0, 1, 2, 3, 4, 5, or 6 oranges. And now the number of bananas is fixed.

If you chose to put in 2 apples you could choose
0, 1, 2, 3, 4, or 5 oranges. And now the number of bananas is fixed.

Do you see the pattern? How many total choices do you have?
 
  • #6
In any case, I don't think the formula is C(n+k-1, k). Perhaps it is C(n+k-1, k-1)? The general idea behind this is an alternative way of looking at the problem. Suppose apples are represented by A, bananas by B, oranges by O. Then, let apples always come before bananas always come before oranges, so a basket of 3 apples, 1 banana, and 3 oranges would be the string AAABOOO. Now, an empty basket would be just seven blank spaces waiting for letters to be put in, so: _ _ _ _ _ _ _

To this string of seven letters, we add two separators. Thus, we have _ _ _ _ _ _ _ | |

If we put these nine symbols in any distinguishable order, we will have a distinct fruit basket, if we keep in mind that anything to the left of the first separator is an apple, anything in between the two is a banana, and anything after the second separator is an orange.

For example, the arrangement for 3 apples, 1 banana, and 3 oranges is now: _ _ _ | _ | _ _ _

Basically, we have 9 places (this is n+(k-1)) where k-1 is the number of separators. From that, we choose 2 places (this is k-1) for the separators, and order doesn't matter. Thus, there should be C(n+k-1, k-1) arrangements.

Hopefully, that's right.
 

What is the difference between combination and permutation?

Combination and permutation are both ways of arranging objects, but the key difference between the two is that combination is an arrangement where the order of the objects does not matter, while permutation is an arrangement where the order of the objects does matter.

How do I calculate the number of combinations?

The number of combinations can be calculated using the formula nCr = n! / r!(n-r)!, where n is the total number of objects and r is the number of objects being chosen. For example, if you have 5 objects and you want to know the number of ways you can choose 3 of them, the calculation would be 5C3 = 5! / 3!(5-3)! = 10.

How do I calculate the number of permutations?

The number of permutations can be calculated using the formula nPr = n! / (n-r)!, where n is the total number of objects and r is the number of objects being arranged. For example, if you have 5 objects and you want to know the number of ways you can arrange 3 of them, the calculation would be 5P3 = 5! / (5-3)! = 60.

What is the use of combination and permutation in real life?

Combination and permutation are used in various fields such as mathematics, computer science, statistics, and finance. In real life, they can be used to solve problems related to probability, counting, and optimization. For example, combination can be used to calculate the probability of winning a lottery game, and permutation can be used to generate unique passwords for online accounts.

What are some common mistakes to avoid when solving combination/permutation problems?

One common mistake is mixing up the formulas for combination and permutation, which can lead to incorrect calculations. It is also important to carefully consider the given conditions and restrictions in the problem, as they can affect the calculation. Another mistake to avoid is forgetting to account for repeating objects, if present, in the calculation.

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