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Zachary wants to buy four ice­creams ­ one each of four different bran

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Zachary wants to buy four ice­creams ­ one each of four different bran  [#permalink]

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New post 08 Nov 2017, 22:57
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Question Stats:

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Zachary wants to buy four ice­creams ­ one each of four different brands. If each of the four brands serve three different flavors ­ chocolate, vanilla, and strawberry, how many different combinations of ice­creams can he buy, provided that he doesn’t end up buying all ice­creams of the same flavor?

A. 60
B. 78
C. 126
D. 156
E. 256

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Re: Zachary wants to buy four ice­creams ­ one each of four different bran  [#permalink]

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New post 09 Nov 2017, 06:49
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nkmungila wrote:
Zachary wants to buy four ice­creams ­ one each of four different brands. If each of the four brands serve three different flavors ­ chocolate, vanilla, and strawberry, how many different combinations of ice­creams can he buy, provided that he doesn’t end up buying all ice­creams of the same flavor?

A. 60
B. 78
C. 126
D. 156
E. 256



hi..

four different brands and each having 3 flavours..
total ways to choose 4 with the given restriction that one of each brand is chosen = \(3*3*3*3 = 81\)

ways in which all flavours are same = 1*1*1*1*3= 3, since there are three flavours

so answer = \(81-3=78\)

B
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Re: Zachary wants to buy four ice­creams ­ one each of four different bran  [#permalink]

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New post 10 Oct 2018, 07:05
chetan2u wrote:
nkmungila wrote:
Zachary wants to buy four ice­creams ­ one each of four different brands. If each of the four brands serve three different flavors ­ chocolate, vanilla, and strawberry, how many different combinations of ice­creams can he buy, provided that he doesn’t end up buying all ice­creams of the same flavor?

A. 60
B. 78
C. 126
D. 156
E. 256



hi..

four different brands and each having 3 flavours..
total ways to choose 4 with the given restriction that one of each brand is chosen = \(3*3*3*3 = 81\)

ways in which all flavours are same = 1*1*1*1*3= 3, since there are three flavours

so answer = \(81-3=78\)

B


Can you please explain the highlighted portion?
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Re: Zachary wants to buy four ice­creams ­ one each of four different bran  [#permalink]

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New post 13 Oct 2018, 17:01
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nkmungila wrote:
Zachary wants to buy four ice­creams ­ one each of four different brands. If each of the four brands serve three different flavors ­ chocolate, vanilla, and strawberry, how many different combinations of ice­creams can he buy, provided that he doesn’t end up buying all ice­creams of the same flavor?

A. 60
B. 78
C. 126
D. 156
E. 256


Since each of the 4 brands has 3 flavors, the number of different combinations of ice cream he can buy is 3^4 = 81 if there are no restrictions. However, since he can’t buy all ice cream of the same flavor, that is, he can’t buy all 4 chocolate, all 4 vanilla, and all 4 strawberry, we must deduct 3 from 81. So he still can buy 81 - 3 = 78 different combinations of ice cream.

Answer: B
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Re: Zachary wants to buy four ice­creams ­ one each of four different bran  [#permalink]

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New post 14 Oct 2018, 08:06
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ian32666 wrote:
chetan2u wrote:
nkmungila wrote:
Zachary wants to buy four ice­creams ­ one each of four different brands. If each of the four brands serve three different flavors ­ chocolate, vanilla, and strawberry, how many different combinations of ice­creams can he buy, provided that he doesn’t end up buying all ice­creams of the same flavor?

A. 60
B. 78
C. 126
D. 156
E. 256



hi..

four different brands and each having 3 flavours..
total ways to choose 4 with the given restriction that one of each brand is chosen = \(3*3*3*3 = 81\)

ways in which all flavours are same = 1*1*1*1*3= 3, since there are three flavours

so answer = \(81-3=78\)

B


Can you please explain the highlighted portion?


Hi,
Your query can be easily clarified using the slot method, which I have explained below.

The number of different combinations of ice­creams that he can buy, provided that he doesn’t end up buying all ice­creams of the same flavor = (Total number of combinations without any restriction) minus (the number of combinations where the ice creams are of same flavor.)

Using slot method, total number of icecreams can be obtained by one slot for each of the 4 icecreams as _*_*_*_ . Since three flavors are possible in each of these slots (four separate brands with three flavors each) the slots become 3*3*3*3 = 81

The number of combinations where the ice creams are of same flavor can be obtained again by 4 slots as used previously, which will be _*_*_*_ . Here, please note that, the first slot has the choice of 3 ice creams, whereas the following slots do not have such a choice and MUST be of the same flavor that was previously chosen in the first slot, whatever the flavor might be. This can be represented in the slot as 3*1*1*1 which equals 3. As mentioned above, the second, third and fourth slots have 1 in them because they do not have the choice of three flavors and are forced to follow whatever is the first icecream, while the first slot has 3 because it has the choice of 3 flavors.

And hence, 81 minus 3 gives your 78 which is OA.

Hope my explanation helps you at least in the slightest bit! :-D
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Re: Zachary wants to buy four ice­creams ­ one each of four different bran &nbs [#permalink] 14 Oct 2018, 08:06
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