Ingots, cars, ranking and short-list

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George Bliss distributes his proceeds to his $18$ associates.

• (a) He distributes $15$ gold bars, all of different weight (he may give them all to the same person): in how many ways?
• (b) He distributes $5$ identical Lamborghini cars to his best associates, no more than one each: in how many ways?
• (c) He ranks all $18$ associates, from most to least trustworthy: how many rankings?
• (d) He extracts a short-list of the top $6$, ordered from most to least trustworthy: how many mini-rankings?

Solution

(a) Each (distinguishable) bar goes to one of the $18$ associates: arrangements with repetition: $18^{15}=6\,746\,640\,616\,477\,458\,432.$ (b) Identical cars, at most one each: choose the $5$ associates out of $18$: $\dbinom{18}{5}=8\,568$. (c) Ordering of all $18$ associates: permutations $18!=6\,402\,373\,705\,728\,000$. (d) Ordered ranking of $6$ associates out of $18$: simple arrangements $D_{18,6}=18\cdot17\cdot16\cdot15\cdot14\cdot13=13\,366\,080$. $\boxed{\ a)\ 18^{15};\quad b)\ 8\,568;\quad c)\ 18!;\quad d)\ 13\,366\,080\ }$