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.
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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
For those who aren't familiar with the formula, let me explain in brief why the formula works!
When we are considering n ! ways, note that we're also permuting the identical items within the n ! arrangements. But permuting identical items result in getting the same arrangement which shouldn't be counted again. If there are x types of items with r x identical items of type x , then while permuting, we originally counted each arrangement as x ∏ ( r i ) ! arrangements because we permuted the identical r i items in ( 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 ! arrangements, the actual number of arrangements is x ∏ ( r i ) ! n ! by unitary method.
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total fruit= 1 9
number of ways to eat those fruit = 3 ! × 2 ! × 4 ! × 1 0 ! 1 9 !
= 1 1 6 3 9 6 2 8 0