In a shop there are five types of ice creams available. A child can buy six ice creams in how many ways?

(actual answer is 10c4. but why isnt the answer 5 power 6?)

Hi Saideep,
Let the 5 types of ice cream be a,b,c,d,e.
Now the child has to be 6 ice creams.
So, lets say there are 6 blanks like _ _ _ _ _ _.
Each of the blank can be filled in 5 ways (from the 5 tyes of ice cream).
Please note that here we are allowed to have repetition.
Like for example the child could have taken: a,b,c,a,a,b. (So there can be a total of 5*5*5*5*5*5 = 5⁶ different combinations of ice cream the child could have bought).

So no of ways = 5⁶.
Best Regards,
Ashwin (IIT Madras).

The answer provided by you is wrong. The count of 56 selects and arranges 6 ice-creams in an order after selecting from 5 different ice-creams. But we want to get no of ways by which the child can buy 6 ice-creams and NOT arrange them.
The correct approach can be as follows:
Let no of ice-creams bought of each type as a,b,c,d,e . So,
a+b+c+d+e=6
as the boy needs 6 ice-creams, the answer is 10C4.