Counting

Inside a dusty chest marked “Halloween costumes” in the family attic, there are four different outfits (a wizard's cape, army fatigues, and two others), five different headgears (a batman helmet, a headband, a tiara, etc.), and nine different accessories (a wand, a lightsaber, a pipe, and many others). If a child were to choose a costume by selecting one outfit, one headgear, and one accessory, ho…

What if the child were permitted to skip one or more of the items (for instance, choosing a costume with an outfit and accessory, but no headgear)?

Go back to when the child did have to choose something from each category, but now say they can have any number of accessories (so they could have the wizard's cape, a batman helmet, plus a lightsaber, pipe, and scepter). Now how many costumes are there?

Okay, that's overkill. A kid only has two hands, after all, so handling nine accessories would be a dextrous challenge. Let's say instead that a child can choose up to three accessories (but must have at least one). Now how many costume choices are there?

When it's finally time to go trick-or-treating, we join up with our next-door neighbors and split up the families into somewhat haphazard groups. There are eleven total children, and six adults. Now let's say each group must have between 3 and 5 members, and must have at least one adult (to stay safe) and at least one kid (otherwise what's the point?) How many different groups are possible?

To encourage rivalry and gluttony, we're going to give a special certificate to the child who collects the most candy at the end of the night. And while we're at it, we'll give 2nd-place and 3rd-place certificates as well. How many different ways could our 1st-2nd-3rd contest turn out?

Finally, what if we want every kid to get a certificate with their name and place-of-finish on it. How many possibilities? (Assume no ties.)