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Combination Question

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Combination Question

by YavuzSelim111 » Fri Feb 21, 2014 7:09 pm

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Hi there, I am having difficulty in trying to under how to work this problem. Any help would be appreciated. Thanks in advance:

Jane must select three different items for each dinner she will serve. The items are to be chosen from among 5 different vegetarian and 4 different meat selections. If at least one of the selections must be vegetarian, how many different dinners could Jane create?

A 30
B 40
C 60
D 70
E 80

What I did: Total Possible Dinners choices - Only Meat Selections = At least one selection veg
= 9!/6! - 4!/1! = 9*8*7 - 4*3*2 = At least one selection is veg = 21, but 21 is not even in the answer choices. Where am I going wrong?
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by Patrick_GMATFix » Fri Feb 21, 2014 7:41 pm
Your logic is right. # of dinners with at least one vegetarian dish = Total - # with no vegetarian dish.

Your mistake is that you used the permutation rather than combination formula. Since there no distinction is made once the 3 items are selected, the order in which they are selected is irrelevant and the combination formulas should be used.

Total # of dinners possible = 9!/(3!6!) = (9*8*7)/(3*2*1) = 84
Total # of ways to pick only meat dinners = 4!/(3!1!) = 4 (this should make sense; the number of ways to pick 3 options from 4 = # of ways to leave out 1 option = 4 ways)

# of dinners with at least 1 vegetarian dish = 84 - 4 = 80

-Patrick
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by YavuzSelim111 » Sat Feb 22, 2014 5:47 pm
Thank you.
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by Patrick_GMATFix » Sat Feb 22, 2014 8:02 pm
Don't mention it :)
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by theCodeToGMAT » Sun Feb 23, 2014 12:21 am
Alternative Approach

9c3 - 4c3

9x8x7/6 - 4

3x4x7 - 4

84 - 4

80
R A H U L
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Re: Combination Question

by gmatstudent2017 » Sun Mar 01, 2020 12:24 pm
Hi, my approach to this question was a little different. Can you help me identify where I went wrong please?

Since there are three dishes Jane has to pick of which one has to be vegetarian I used the slot method.

The first combination is veggie, meat, meat. There are 5 veggie dishes she can pick from so the first slot is 5. Then there are 4 meat dishes she can pick from so the second slot is 4. Lastly, there are only 3 meat dishes left that she can pick the second dish from so the third slot is 3. This gives a total of 60.

__5__ * __4__ * __3__

Then the other combination is veggie, veggie, meat which gives you a total of 80.

__5__ * __4__ * __4__

The third combination is veggie, veggie, veggie which gives you a total of 60.

__5__ * __4__ * __3__

60+80+60 = 200 which isn't even an option. Can someone tell me where I went wrong?

Thank you!
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