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Amika wants to buy 4 cupcakes from an infinite supply of three types of cupcakes: chocolate, vanilla & gems-laden. In how many different ways can she buy cupcakes, if :

(1) order of buying the cupcakes is important, and

(2) arbitrary order is permitted

vidyarthi
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rohit
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    Do you care what order she buys them in? What have you tried? Where are you stuck? For a problem this small you could try to list the possibilities by hand. – Ross Millikan Nov 18 '16 at 04:54
  • Can it then be 3^4=81 ways?? –  Nov 18 '16 at 05:02
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    @Rohan: that would be correct if the order matters. I suspect it is not supposed to matter, that getting choc-choc-van-van is the same as getting van-choc-choc-van, but if we don't know it is hard to answer the question. – Ross Millikan Nov 18 '16 at 05:18
  • Gem-laden? What, is she buying cupcakes for a certain purple dragon in Equestria? I'm wondering if this is just an unusual translation error and it was intended to be "sprinkles" instead or something. – JMoravitz Nov 18 '16 at 05:23

2 Answers2

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Suppose order is important. first cupcake $3$ ways, second $3$ ways, third in $3$ ways, fourth in $3$ ways, So, $3^4$ in total

if order is not important, it is $\dbinom{3+4-1}{4}$
(this is similar to the stars and bars pattern I guess)

Kiran
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The number of different ways she can buy 4 cupcakes is $3^4=81$ ways because she can choose arbitrarily any cupcake from $3$ varieties. But, if order of choosing is unimportant, then it is $\binom{3+4-1}{3}$ which equals $15$. The formula in the general case is taken from combination with repititions

Community
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vidyarthi
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