A Certain Ice-cream store has $31$ flavours of ice-creams available. In how many ways can we order a dozen ice-cream cones if
$a)$ we don't want the same flavor more than once?
$b)$ a flavor may be ordered as many as $12$ times?
$c)$ a flavor may be ordered not more than $11$ times?
I think if there are no restrictions then it will be C(12+31-1, 31) but I dont know how to solve this when there are restrictions like this question.Thank you
Best Answer
a) $\dfrac{31!}{(31-12)!12!}=141120525$
b) $\dfrac{(31+12-1)!}{(31-1)!12!}=11058116888$
c) first take alle possibble combinations, so 11058116888.0, then take all 31 combinations which do have more than 11 of the same flavor and subtract: $11058116888-31=11058116857$