Suppose that we have pn objects and we have p slots, how many ways can we distribute those pn objects on those p slots so that each slot has exactly n object?

basically the answer should be (pn)! /(n! ^p)

but i have been trying to prove this and i just can't,

since we are distributing pn objects on p slots (keep in mind that the slots are distinguishable but the position within the slots are not) we first need to chose a slot, and then put n objects in it, and do it for all p slots

which means the answer should be c(p,1)*c(pn, n)*c(p,2)*c(n(p-1))**c(p, p)*c(n,2n),i am not completely sure on this last one but if i simplify this last one i get (p! *(/(n! ^p) and that's wrong so where is my mistake?

Muscles, a membership-only gym is hoping to open a new branch in a small city in pennsylvania that currently has no fitness centers. according to their research approximately 12,600 residents live within driving distance of the gym. muscles sends out surveys to a sample of 300 randomly selected residents in this area (all of who respond) and finds that 40 residents say they would visit a gym if one was located in their area. based on the past survey research, muscles estimates that approximately 30% of these respondents would actually join the gym if they opened one in the area. based on this information and the results of the sample survey, about how many residents should muscles expect to join its new branch? a) 134 b) 504 c) 1,680 d) 3,780

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