Question Number 27514 by NECx last updated on 08/Jan/18
$${The}\:{president}\:{of}\:{a}\:{republic}\:{wishes} \\ $$$${to}\:{assign}\:{the}\:{premiership},{the}\: \\ $$$${vice}-{premiership},{and}\:{six}\:{other} \\ $$$${cabinet}\:{posts}\:{to}\:{a}\:{selected}\:{group} \\ $$$${of}\:\mathrm{8},{comprising}\:{of}\:\mathrm{6}\:{career} \\ $$$${diplomats}\:{and}\:\mathrm{2}\:{business}\:{men}.{If} \\ $$$${none}\:{of}\:{the}\:{business}\:{men}\:{can}\:{be} \\ $$$${made}\:{premier}\:{or}\:{vice}\:{premier},{in} \\ $$$${how}\:{many}\:{ways}\:{can}\:{the}\:\mathrm{8}\:{posts} \\ $$$${be}\:{assigned}\:{to}\:{the}\:\mathrm{8}\:{people}? \\ $$
Answered by Rasheed.Sindhi last updated on 08/Jan/18
$$\mathrm{The}\:\mathrm{upper}\:\mathrm{2}\:\mathrm{posts}\:\mathrm{can}\:\mathrm{be}\:\mathrm{selected} \\ $$$$\mathrm{from}\:\mathrm{only}\:\mathrm{6}\:\mathrm{diplomats} \\ $$$$\mathrm{Hence}\:\mathrm{they}\:\mathrm{can}\:\mathrm{be}\:\mathrm{selected} \\ $$$$\mathrm{in}\:\mathrm{6}×\mathrm{5}=\mathrm{30}\:\mathrm{ways}\left(\mathrm{One}\:\mathrm{can}\:\mathrm{be}\right. \\ $$$$\mathrm{selected}\:\mathrm{in}\:\mathrm{6}\:\mathrm{ways}\:\mathrm{and}\:\mathrm{after}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{other}\:\mathrm{can}\:\mathrm{be}\:\mathrm{selected}\:\mathrm{from} \\ $$$$\left.\mathrm{5}\:\mathrm{remaining}\:\mathrm{diplomates}\right) \\ $$$$\mathrm{After}\:\mathrm{the}\:\mathrm{above}\:\mathrm{selection}\:\mathrm{there} \\ $$$$\mathrm{remains}\:\mathrm{6}\:\mathrm{persons}\:\left(\mathrm{4}\:\mathrm{diplomates}\right. \\ $$$$\left.\mathrm{and}\:\mathrm{2}\:\mathrm{business}\:\mathrm{men}\right)\:\mathrm{and}\:\mathrm{6}\:\mathrm{posts} \\ $$$$\mathrm{so}\:\mathrm{they}\:\mathrm{can}\:\mathrm{be}\:\mathrm{filled}\:\mathrm{6}!\:\mathrm{ways}. \\ $$$$\therefore\:\mathrm{Toal}\:\mathrm{ways}=\left(\mathrm{6}.\mathrm{5}\right)\left(\mathrm{6}!\right)=\mathrm{21600} \\ $$
Commented by NECx last updated on 29/Apr/18
$${Thank}\:{you}\:{so}\:{much}\:{sir}. \\ $$