Previous in Permutation and Combination Next in Permutation and Combination
Question Number 224261 by Tawa11 last updated on 30/Aug/25
If I have 10 people and I want to create pairings, so that every person is matched with every other person at least once, how many unique pairings will there be in total?
Commented by mr W last updated on 31/Aug/25
$${no}\:{clear}\:{enough}\:{what}\:{you}\:{are}\:{asking}. \\ $$$${do}\:{you}\:{ask}\:{how}\:{many}\:{ways}\:{to}\:{select} \\ $$$${a}\:{pair}\:{from}\:\mathrm{10}\:{people}? \\ $$$${then}\:{answer}\:{is}\:{C}_{\mathrm{2}} ^{\mathrm{10}} =\mathrm{45}\:{ways}. \\ $$$$ \\ $$$${or}\:{do}\:{you}\:{ask}\:{how}\:{many}\:{ways}\:{to}\: \\ $$$${buid}\:\mathrm{5}\:{pairs}\:{from}\:\mathrm{10}\:{people}? \\ $$$${then}\:{answer}\:{is}\:\frac{\mathrm{10}!}{\left(\mathrm{2}!\right)^{\mathrm{5}} ×\mathrm{5}!}=\mathrm{945}\:{ways}. \\ $$
Commented by TonyCWX last updated on 31/Aug/25
$${You}'{re}\:{missing}\:{one}\:{person}. \\ $$$${Notice}\:{that}\:{it}'{s}\:``\mathrm{10}\:{people}\:\boldsymbol{{and}}\:\boldsymbol{{I}}\:''. \\ $$
Commented by mr W last updated on 31/Aug/25
$${Sorry},\:{my}\:{english}\:{is}\:{not}\:{so}\:{good}. \\ $$$${When}\:{you}\:{say}\:``\underline{{I}\:{have}\:\mathrm{3}\:{dogs}\:{and}\:{I}} \\ $$$$\underline{{want}\:{to}\:{feed}\:{them}.}''\:{According}\:{to} \\ $$$${my}\:{understanding},\:{you}\:{just}\:{feed} \\ $$$${the}\:\mathrm{3}\:{dogs},\:{not}\:{including}\:{yourself}.\: \\ $$$${But}\:{according}\:{to}\:{you},\:\mathrm{4}\:{persons}\: \\ $$$${should}\:{be}\:{fed},\:{because}\:{one}\:{persion}\: \\ $$$${is}\:{missing},\:{since}\:{it}'{s}\:``\mathrm{3}\:{dogs}\:\boldsymbol{{and}}\:\boldsymbol{{I}}''. \\ $$$${please}\:{recheck}! \\ $$
Commented by fantastic last updated on 31/Aug/25
$${The}\:{language}\:{of}\:{question}\: \\ $$$${should}\:{not}\:{be}\:{misleading} \\ $$$$ \\ $$
Commented by Tawa11 last updated on 31/Aug/25
If I have 10 people and want to create pairings where every person is matched with every other person at least once, how many unique pairings will there be?"
Commented by Tawa11 last updated on 31/Aug/25
$$\mathrm{Sir}\:\mathrm{mrW}, \\ $$$$\mathrm{with}\:\mathrm{what}\:\mathrm{you}\:\mathrm{stated},\:\mathrm{the}\:\:\overset{\mathrm{10}} {\:}\mathrm{C}_{\mathrm{2}} \:\:=\:\:\mathrm{45} \\ $$$$\mathrm{is}\:\mathrm{correct}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{want}\:\mathrm{to}\:\mathrm{understand}\:\mathrm{the} \\ $$$$\mathrm{grammar}\:\mathrm{too},\:\mathrm{that}\:\mathrm{is}\:\mathrm{why}\:\mathrm{I}\:\mathrm{sent}\:\mathrm{it}\:\mathrm{here}. \\ $$
Commented by TonyCWX last updated on 01/Sep/25
$${Sorry}\:{sir}. \\ $$$${My}\:{understanding}\:{was}\:{bad}\:{this}\:{time}. \\ $$