Four combinatorial counts with kings, dogs, nuggets and a padlock

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• (a) The royal family of England marries each of its three sons to three of the four daughters of the king of France. In how many ways can the three sons be matched with the four daughters?
• (b) Count Dragula feeds his three dogs with seven identical bones (a dog may get several bones, or none), using all seven bones. In how many ways can he distribute them?
• (c) John Piccon’s bag contains ten different nuggets; he gives three to his son. In how many ways can the three nuggets be chosen?
• (d) A padlock has three dials, each with five symbols (ship, caiman, shark, octopus, pirate); symbols may repeat. How many different combinations does the padlock have?

Solution

(a) Assign $3$ distinct daughters to the $3$ distinct sons: order matters and no repetition, so simple arrangements of $4$ objects taken $3$ at a time: $D_{4,3}=4\cdot 3\cdot 2=24.$

(b) Seven identical bones split among $3$ dogs (possibly $0$): this is combinations with repetition (stars and bars), i.e. placing $2$ separators among $7$ bones: $C^{\,r}_{3,7}=\binom{7+3-1}{3-1}=\binom{9}{2}=36.$

(c) Three distinct nuggets, order irrelevant: simple combinations $\binom{10}{3}=120.$

(d) Three independent dials, $5$ symbols each with repetition: arrangements with repetition $5^3=125.$

$\boxed{(a)\,24;\quad (b)\,36;\quad (c)\,120;\quad (d)\,125}$