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Question-115916




Question Number 115916 by I want to learn more last updated on 29/Sep/20
Answered by mr W last updated on 30/Sep/20
say the first shelf has k books,  then the second shelf has 15−k books.  1≤k≤14  to select k books there are C_k ^(15)  ways.  to place k books there are k! ways.  to place 15−k books there are (15−k)! ways.    totally:  Σ_(k=1) ^(14) C_k ^(15) k!(15−k)!  =Σ_(k=1) ^(14) 15!  =14×15!
$${say}\:{the}\:{first}\:{shelf}\:{has}\:{k}\:{books}, \\ $$$${then}\:{the}\:{second}\:{shelf}\:{has}\:\mathrm{15}−{k}\:{books}. \\ $$$$\mathrm{1}\leqslant{k}\leqslant\mathrm{14} \\ $$$${to}\:{select}\:{k}\:{books}\:{there}\:{are}\:{C}_{{k}} ^{\mathrm{15}} \:{ways}. \\ $$$${to}\:{place}\:{k}\:{books}\:{there}\:{are}\:{k}!\:{ways}. \\ $$$${to}\:{place}\:\mathrm{15}−{k}\:{books}\:{there}\:{are}\:\left(\mathrm{15}−{k}\right)!\:{ways}. \\ $$$$ \\ $$$${totally}: \\ $$$$\underset{{k}=\mathrm{1}} {\overset{\mathrm{14}} {\sum}}{C}_{{k}} ^{\mathrm{15}} {k}!\left(\mathrm{15}−{k}\right)! \\ $$$$=\underset{{k}=\mathrm{1}} {\overset{\mathrm{14}} {\sum}}\mathrm{15}! \\ $$$$=\mathrm{14}×\mathrm{15}! \\ $$
Commented by I want to learn more last updated on 30/Sep/20
Thanks sir, but the text answer says   14 × 15!
$$\mathrm{Thanks}\:\mathrm{sir},\:\mathrm{but}\:\mathrm{the}\:\mathrm{text}\:\mathrm{answer}\:\mathrm{says}\:\:\:\mathrm{14}\:×\:\mathrm{15}! \\ $$
Commented by mr W last updated on 30/Sep/20
i considered my answer again.
$${i}\:{considered}\:{my}\:{answer}\:{again}.\: \\ $$
Commented by I want to learn more last updated on 01/Oct/20
Thank you sir for your time
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir}\:\mathrm{for}\:\mathrm{your}\:\mathrm{time} \\ $$

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