From 3 apples, 4 plums and 2 grapefruits how many selections

Question

From 3 apples, 4 plums and 2 grapefruits how many selections can be made taking atleast one of each kind?

Please let me know the concept behind this.

Thanks!!

banerjeea_98 · Answer

assumimg fruits r considered diff....

pick 1 apple = 3C1 = 3 ways
2 apples....= 3C2 = 3 ways
3 apples = 3C3 = 1 ways

total ways for apple = 3+3+1 = 7 ways


pick 1 plum = 4C1 = 4 ways
2 plum ....= 4C2 = 6 ways
3 plum = 4C3 = 4 ways
4 plum = 4C4 = 1 ways

total ways for plum= 4+6+4+1 = 15 ways

pick 1 grapefruit = 2C1 = 2 ways
2 grapefruit ....= 2C2 = 1 ways


total ways for grapefruit = 3 ways


Total ways = 7*15*3 = 315

gmat2me2 · Answer

Baner that is one way of doing it but in the book I was doing from he gave a very simple step (which I didnt understand)

(2^3 -1)(2^4-1)(2^2-1)=315

shishirgupta77 · Answer

Think of it this way:

How many ways you can choose from 3 apples?

2^3=8

atleast one apple has to be selected, which means (2^3-1). Similarly apply for other.

Thanks
Shishir

HongHu · Answer

The notion 2^n is for the Yes/No answer. For each of the three apples, there are two choices: You can choose it (Y) or not choose it (N). So the total outcome for three apples would be 2^3. Now, since you have to choose at least one apple, this means you have to take out the one outcome where no apple is chosen (NNN). Therefore you get 2^3-1. Do the same thing for other fruits.

When the number of fruits for each type is not too big, baner's approach may be more straight forward.

Rupstar · Answer

How r we doing this without knowing the total number of selections to be made....im confused

ketanm · Answer

The question itself is to find the total number of selections. One of the selection with minimum number of fruits could be to select one from each category. Thus minimum fruits that can be choosen are 3 and maximum fruits that can be chosen are 9. So the number of fruits that can be choosen start from 3 upto 9. For each number the question need to know, how many ways can this selection be made. Example, 3 fruits can only be choosen in one way i.e 1 apple, 1 plum and 1 grapefruit. Similarly 4 fruits can be choosen in many ways 2 apples, 1 plums and 1 grapefruits 1 apples, 2 plums and 1 grapefruits 1 apples, 1 plums and 2 grapefruits And so on..... Hope I have not confused you further... :-)

Ketan

Rupstar · Answer

ok now it makes sense
thanks mate!

gmat2me2 · Answer

HongHu:

Thanks !!! Thats it I could now get the concept. 

Baner : your process of evaluation is right as always but I didnt know why this 2^n concept is being used.......

Bionomial Theorem says

nC0 + nC1 + nC2 +...nCn = 2^n

So the total # of combinations of 'n' things (taken in portions/all) = 2^n

vprabhala · Answer

this makes my concept clear..thanks honghu.

From 3 apples, 4 plums and 2 grapefruits how many selections

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