Mango Mania

We have 7 mangoes, to be divided into 4 different types. This is a direct application of sticks and stones, then: we have 7 stones and 3 sticks, giving a total of $\dbinom{7+3}{3}=\dbinom{10}{3}=\boxed{120}$ .

Not a bad problem, although I would encourage you try to make your problems a little deeper. Keep on trying!

number of non-integral solution , a + b + c + d = 7 , C(10,3) = 120

The following i write is visual form of what Daniel Liu wrote , hope it can be help of those who absorbs maths visually and also those who didn't learn the writing method of it or is trying to sum it manually.

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given dat we have mango type a , b , c , d and any 7 mangoes will weight 1kg

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if she pick only one type of mango - we have 4 patterns

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a x 7

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b x 7

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c x 7

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d x 7

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if she pick 2 types of mango - we got 6 pattern

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ab , ac , ad bc , bd cd

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now pick any type of the 6 pattern and do stick and stone method. In this case i choose a and b.

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_ _ _ _ _ _ S _ = 6a + 1b = 7 mangoes

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_ _ _ _ _ S _ _ = 5a + 2b = 7 mangoes

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_ S _ _ _ _ _ _ = 1a + 6b = 7 mangoes

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Now take a look at the capital letter 'S' which means the stick. Left side of S is all type a mango and right side is type b.

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Of all the 7 stones(mango) , there is 6 interval where you can put the stick into as shown above. Since you can shift the stick 6 times , you will have 6 combinations as shown above.

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now don't forget that we got 6 patterns of fruits

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ab , ac , ad bc , bd cd

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each pattern has 6 combinations which means 6x6 = 36. 36 ways to buy 2 types of mangoes which weight 1kg

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*beyond this point you might need calculator or you can just type it in this online math program http://www.wolframalpha.com/

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3 types of fruit - 4 patterns or 4c3(4 choose 3) in calculator or manual list - abc , abd , acd , bcd

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_ _ _ _ _ S _ S _ = 5a + 1b + 1c = 7 mangoes

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now we got 2 sticks to put among the 6 intervals. 6c2 = 15 4 pattern x 15 combination = 60

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4 types of fruit - 1 pattern

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_ _ _ _ S _ S _ S _

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3 Sticks to put among the 6 interval. 6c3 = 20

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Total buying pattern from single fruit all the way to 4 types of fruit = 4 + 36 + 60 + 20 = 120

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presto you go it , however please take note that visual method may only help when the selection is less. if it goes all the way from a-z its better to use writing method as shown by Daniel Liu.

Here is the gf you ought to expand:

$\frac{1}{(1-x)^4}$

Expand its taylor series and look for the x^7 term

I know the result but I do not know how to get it.

Gauss

If Gauss himself cannot explain how he does stuff, how can I?