Agents, rifles, ingots and cars

Text

Polite Ninja sends a contingent of $17$ secret agents to guard the $13$ (distinct) hotels of the capital, so that each hotel is assigned one agent, and no more than one.

• (a) In how many ways can he assign the agents to the hotels?
• (b) He wants to equip the $17$ agents with $10$ identical sniper rifles, no more than one each: in how many ways can he hand them out?
• (c) He distributes $40$ identical gold ingots to his agents, possibly more than one each: in how many ways?
• (d) He posts a list of $10$ different car models and asks each agent to mark their favourite (the same model may be chosen by several agents): in how many ways can the choice happen?

Solution

(a) Each of the $13$ hotels gets a distinct agent: simple arrangements of $17$ objects in class $13$: $D_{17,13}=\frac{17!}{4!}=17\cdot16\cdots5=14\,820\,309\,504\,000.$ (b) The rifles are identical and at most one each: just choose the $10$ armed agents out of $17$ — simple combinations: $\binom{17}{10}=\binom{17}{7}=19\,448.$ (c) Identical ingots, no limit: combinations with repetition ($40$ objects into $17$ “urns”, i.e. stars and bars): $\binom{40+17-1}{17-1}=\binom{56}{16}=41\,648\,951\,840\,265.$ (d) Each agent independently picks $1$ model out of $10$: arrangements with repetition: $10^{17}=100\,000\,000\,000\,000\,000.$ $\boxed{\ a)\ 14\,820\,309\,504\,000;\quad b)\ 19\,448;\quad c)\ 41\,648\,951\,840\,265;\quad d)\ 10^{17}\ }$