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 flavoursso 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!