Fruit Mania

Probability Level 3

In how many different ways can I eat 3 bananas, 2 oranges, 4 plums and 10 grapes?

Details and Assumptions :

  • All fruits of the same type are identical.

  • I have to eat all the fruit.

  • I can only eat one piece of fruit at once.

Image Credit: Flickr France-♥


The answer is 116396280.

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Melissa Quail by Melissa Quail , 21, United Kingdom

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2 solutions

Samuel Ayinde
Apr 1, 2015

total fruit= 19 19

number of ways to eat those fruit = 19 ! 3 ! × 2 ! × 4 ! × 10 ! \frac{19!}{3!\times2!\times4!\times10!}

= 116396280 116396280

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Aquib Immanuel
Mar 31, 2015

If A out of N items are identical, then the number of different permutations of the N items is N!/A! If a set of N items contains A identical items, B identical items, and C identical items etc.., then the total number of different permutations of N objects is N!/(A!⋅B!⋅C!...!) here the answer would be

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For those who aren't familiar with the formula, let me explain in brief why the formula works!

When we are considering n ! n! ways, note that we're also permuting the identical items within the n ! n! arrangements. But permuting identical items result in getting the same arrangement which shouldn't be counted again. If there are x x types of items with r x r_x identical items of type x x , then while permuting, we originally counted each arrangement as ∏ x ( r i ) ! \displaystyle\prod_x (r_i)! arrangements because we permuted the identical r i r_i items in ( r i ) ! (r_i)! ways corresponding to each required arrangement and the product notation follows by the rule of product when we are taking all types into account.

Hence, since we counted n ! n! arrangements, the actual number of arrangements is n ! ∏ x ( r i ) ! \dfrac{n!}{\displaystyle\prod_x (r_i)!} by unitary method.

Prasun Biswas - 6 years, 2 months ago

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Nice note Prasun !

Venkata Karthik Bandaru - 6 years, 2 months ago

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