Agents, rifles and ingots (variant)

Text

Polite Ninja sends $7$ agents to guard the $3$ (distinct) hotels of the capital, each hotel one agent and no more than one.

• (a) In how many ways can he make the assignment?
• (b) He distributes $5$ identical rifles to the $7$ agents, no more than one each: in how many ways?
• (c) He distributes $12$ identical gold ingots to his agents, possibly more than one to the same person: in how many ways?

Solution

(a) Simple arrangements of $7$ agents in class $3$: $D_{7,3}=7\cdot6\cdot5=210$. (b) Choose the $5$ armed agents out of $7$: $\dbinom{7}{5}=\dbinom{7}{2}=21$. (c) Combinations with repetition ($12$ ingots into $7$ urns): $\dbinom{12+7-1}{7-1}=\dbinom{18}{6}=18\,564$. $\boxed{\ a)\ 210;\quad b)\ 21;\quad c)\ 18\,564\ }$