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Question:

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Textbook Solution:

$4 + 3 - 1 \choose 3$ * $4 + 6 - 1\choose 6$

$\therefore$ 20 * 84 = 1680

My Solution:

$4 + 3 - 1 \choose 3$ + $4 + 6 - 1\choose 6$

$\therefore$ 20 + 84 = 104

Logical Question:

Why should I use rule of product and not the rule of sum ?

I believe rule of sum is used, because the outcome of distributing the oranges are not dependent on the bananas.

2 Answers 2

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A key deciding point: do you have to choose an arrangement of the bananas AND an arrangement of the oranges, or do you have to choose an arrangement of the bananas OR an arrangement of the oranges?

If you have to choose arrangements for both, you use the product rule. (The set of all possible choices is the cartesian product of the choices for one, and the choices for the other).

If you choose an arrangement from one OR from the other, you use the sum rule. (The set of all possible choices is the sum (disjoint union) of the choices for one and the choices for the other).

Hope this helps!

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    This is helpful. Yes, I need to choose the arrangement of the bananas and oranges, so according to your information, then product rule is used. The key is in the words "and" , "or". What if there is a question that it has not clearly provided the use of the words "and" , "or", is there any other way to notice which rule to use ? – 2017-02-23
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For each way to distribute oranges, there are $x$ ways to distribute bananas, whatever $x$ is. You are correct that they are not dependent, but each way of distributing bananas gives a certain number of options for oranges. Adding them up, and you find you are adding (the number of banana ways) up (the number of orange ways) times.

Does this help?

(I left the quantities out since you clearly already have them.)

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    But we are assuming the bananas and oranges to be the same. Based on your explanation, it is not clear to me why the number of bananas give the options for oranges ? – 2017-02-23
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    @Kourosh Well, we are arranging both bananas and oranges. One of the many ways to do this is to consider the number of ways with just your bananas, then pick one of those ways and distribute the oranges. Each banana way gives the same number of orange ways. Does that help? – 2017-02-23
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    I guess I am starting to pick up what you mean. Do you mean that each time we are distributing the fruits among the 4 kids, we distributing 3 bananas and 6 oranges. Since order does not matter and repetition is allowed, the first kid might get all 3 bananas, and 6 oranges or each kid might get one of each kid. Thus, the arrangement of bananas and oranges are related to each other in this way. Do we agree ? – 2017-02-23
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    That will work, sure. – 2017-02-23
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    Ok, thank you for confirming. – 2017-02-23
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    @Kourosh Sure thing. Thanks for being an attentive asker! Hope to see more of you around. :) – 2017-02-23
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    Thanks for motivation, I love discrete math, because it involves the actual reality of the world inside the world of math. All the possibilities and challenges are what makes it interesting to question about :) – 2017-02-23