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Question Number 198851 by Tawa11 last updated on 25/Oct/23
You want to arrange 17 books on the shelf of a bookstore. The shelf is dedicated to the three Toni Morrison novels published between 1977 and 1987: Song of Solomon, Tar Baby, and Beloved. You have many copies of each, but on the shelf you want an even number of Song of Solomon, at least three copies of Tar Baby, and at most four copies of Beloved. How many different arrangements are possible?
Commented by mr W last updated on 25/Oct/23
$${do}\:{you}\:{have}\:{the}\:{answer}? \\ $$
Commented by mr W last updated on 25/Oct/23
$${nikif}\mathrm{99}\:{sir}:\:{can}\:{you}\:{use}\:{your}\:{fortran} \\ $$$${program}\:{to}\:{calculate}\:{this}? \\ $$
Commented by Tawa11 last updated on 25/Oct/23
$$\mathrm{yes}\:\mathrm{sir} \\ $$
Commented by mr W last updated on 25/Oct/23
$${what}'{s}\:{the}\:{answer}\:{given}?\:{is}\:{there} \\ $$$${also}\:{a}\:{solution}\:{given}? \\ $$$${i}'{m}\:{interested}\:{in}\:{alternative}\:{solutions}. \\ $$
Commented by Tawa11 last updated on 25/Oct/23
$$\mathrm{18},\mathrm{072},\mathrm{242} \\ $$
Commented by Tawa11 last updated on 25/Oct/23
$$\mathrm{yes}\:\mathrm{sir} \\ $$
Commented by Tawa11 last updated on 25/Oct/23
$$\mathrm{Should}\:\mathrm{I}\:\mathrm{send}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{now}\:\mathrm{sir}? \\ $$
Commented by nikif99 last updated on 25/Oct/23
$${if}\:\mathrm{0}\:{books}\:{are}\:{allowed}\:{for}\:``{Song}\:{of}\: \\ $$$${Solomon}''\:{or}\:``{Beloved}''\:{I}\:{found} \\ $$$$\mathrm{18}.\mathrm{072}.\mathrm{242}\:{cases}. \\ $$$${If}\:{not},\:{then}\:{cases}\:{are}\:{limited}\:{to} \\ $$$$\mathrm{18}.\mathrm{003}.\mathrm{510}. \\ $$$${Left}\:\mathrm{3}\:{columns}\:{are}\:{possible}\:{number}\:{of} \\ $$$${copies}\:{selected}\:{of}\:{three}\:{titles},\:{when} \\ $$$$\mathrm{0}\:{books}\:{are}\:{allowed}. \\ $$$${Right}\:\mathrm{3}\:{columns}\:{excempt}\:\mathrm{0}\:{books}. \\ $$$${For}\:{each}\:{horizontal}\:{line},\:{number}\:{of} \\ $$$${arrangements}\:{with}\:{repetitions}\:{is} \\ $$$$\frac{\mathrm{17}!}{{a}!×{b}!×{c}!} \\ $$$$ \\ $$
Commented by nikif99 last updated on 25/Oct/23
Commented by Tawa11 last updated on 25/Oct/23
Commented by mr W last updated on 25/Oct/23
$${are}\:{you}\:{also}\:{familiar}\:{with}\:{generating} \\ $$$${functions}? \\ $$
Commented by Tawa11 last updated on 25/Oct/23
$$\mathrm{No}\:\mathrm{sir}. \\ $$$$\mathrm{I}\:\mathrm{saw}\:\mathrm{the}\:\mathrm{method}\:\mathrm{in}\:\mathrm{your}\:\mathrm{previous}\:\mathrm{solution}\:\mathrm{so}, \\ $$$$\mathrm{I}\:\mathrm{bring}\:\mathrm{the}\:\mathrm{question}\:\mathrm{here}. \\ $$
Answered by mr W last updated on 26/Oct/23
$${once}\:{again}\:{i}'{ll}\:{try}\:{to}\:{use}\:{generating} \\ $$$${functions}. \\ $$$${s}={number}\:{of}\:{books}\:``{Song}\:{of}\:{Solomon}'' \\ $$$${t}={number}\:{of}\:{books}\:``{Tar}\:{Baby}'' \\ $$$${b}={number}\:{of}\:{books}\:``{Beloved}'' \\ $$$${s}\Rightarrow\left({x}^{\mathrm{2}} +{x}^{\mathrm{4}} +{x}^{\mathrm{6}} +...\right) \\ $$$${t}\Rightarrow\left({x}^{\mathrm{3}} +{x}^{\mathrm{4}} +{x}^{\mathrm{5}} +...\right) \\ $$$${b}\Rightarrow\left({x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +{x}^{\mathrm{4}} \right) \\ $$$${s}+{t}+{b}=\mathrm{17} \\ $$$${number}\:{of}\:{arrangements}\:{is}\:{the} \\ $$$${coefficient}\:{of}\:{x}^{\mathrm{17}} \:{term}\:{in}\:{expansion} \\ $$$$\mathrm{17}!\left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}+\frac{{x}^{\mathrm{4}} }{\mathrm{4}!}+\frac{{x}^{\mathrm{6}} }{\mathrm{6}!}+...\right)\left(\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{4}} }{\mathrm{4}!}+\frac{{x}^{\mathrm{5}} }{\mathrm{5}!}+...\right)\left({x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{4}} }{\mathrm{4}!}\right) \\ $$$${which}\:{is}\:\mathrm{18}\:\mathrm{003}\:\mathrm{510}.\: \\ $$$$ \\ $$$${if}\:{no}\:{book}\:{from}\:{S}\:{anf}\:{B}\:{is}\:{also} \\ $$$${allowed},\:{then}\:{the}\:{answer}\:{is}\:{the} \\ $$$${coefficient}\:{of}\:{x}^{\mathrm{17}} \:{term}\:{in} \\ $$$$\mathrm{17}!\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}+\frac{{x}^{\mathrm{4}} }{\mathrm{4}!}+\frac{{x}^{\mathrm{6}} }{\mathrm{6}!}+...\right)\left(\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{4}} }{\mathrm{4}!}+\frac{{x}^{\mathrm{5}} }{\mathrm{5}!}+...\right)\left(\mathrm{1}+{x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{4}} }{\mathrm{4}!}\right) \\ $$$${which}\:{is}\:\mathrm{18}\:\mathrm{072}\:\mathrm{242}. \\ $$
Commented by mr W last updated on 25/Oct/23
Commented by mr W last updated on 25/Oct/23
