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How-many-seven-letter-words-can-be-formed-by-using-the-letters-of-the-word-SUCCESS-so-that-neither-two-C-nor-two-S-are-together-

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Question Number 21917 by Tinkutara last updated on 06/Oct/17
How many seven letter words can be formed by using the letters of the word SUCCESS so that neither two C nor two S are together?
$$\mathrm{How}\:\mathrm{many}\:\mathrm{seven}\:\mathrm{letter}\:\mathrm{words}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{formed}\:\mathrm{by}\:\mathrm{using}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{word} \\ $$$$\mathrm{SUCCESS}\:\mathrm{so}\:\mathrm{that}\:\mathrm{neither}\:\mathrm{two}\:\mathrm{C}\:\mathrm{nor} \\ $$$$\mathrm{two}\:\mathrm{S}\:\mathrm{are}\:\mathrm{together}? \\ $$
Commented by mrW1 last updated on 07/Oct/17
S must be separated by other 4 letters: YXYXYXYXY (X means positions for letters UCCE, Y means possible positions for letter S) to arrange the 4 letters UCCE in positions X there are 4!/2! ways, to place 3 letters S in positions Y there are C_3 ^5 ways ⇒4!/2!×C_3 ^5 (2! since letter C repeats twice) note: in some of these ways the two letters C are next to each other. to find the number of ways where two C are next to each other, we can proceed as above, but treat CC as a single letter. that means YXYXYXY ⇒3!×C_3 ^4 ⇒the number of ways such that neither two S nor two C are next to each other is: 4!/2!×C_3 ^5 −3!×C_3 ^4 =24/2×10−6×4=96
$$\mathrm{S}\:\mathrm{must}\:\mathrm{be}\:\mathrm{separated}\:\mathrm{by}\:\mathrm{other}\:\mathrm{4}\:\mathrm{letters}: \\ $$$$\mathrm{YXYXYXYXY} \\ $$$$\left(\mathrm{X}\:\mathrm{means}\:\mathrm{positions}\:\mathrm{for}\:\mathrm{letters}\:\mathrm{UCCE},\:\right. \\ $$$$\left.\mathrm{Y}\:\mathrm{means}\:\mathrm{possible}\:\mathrm{positions}\:\mathrm{for}\:\mathrm{letter}\:\mathrm{S}\right) \\ $$$$\mathrm{to}\:\mathrm{arrange}\:\mathrm{the}\:\mathrm{4}\:\mathrm{letters}\:\mathrm{UCCE}\:\mathrm{in}\:\mathrm{positions} \\ $$$$\mathrm{X}\:\mathrm{there}\:\mathrm{are}\:\mathrm{4}!/\mathrm{2}!\:\mathrm{ways},\:\mathrm{to}\:\mathrm{place}\:\mathrm{3}\:\mathrm{letters} \\ $$$$\mathrm{S}\:\mathrm{in}\:\mathrm{positions}\:\mathrm{Y}\:\mathrm{there}\:\mathrm{are}\:\mathrm{C}_{\mathrm{3}} ^{\mathrm{5}} \:\mathrm{ways} \\ $$$$\Rightarrow\mathrm{4}!/\mathrm{2}!×\mathrm{C}_{\mathrm{3}} ^{\mathrm{5}} \\ $$$$\left(\mathrm{2}!\:\mathrm{since}\:\mathrm{letter}\:\mathrm{C}\:\mathrm{repeats}\:\mathrm{twice}\right) \\ $$$$ \\ $$$$\mathrm{note}:\:\mathrm{in}\:\mathrm{some}\:\mathrm{of}\:\mathrm{these}\:\mathrm{ways}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{letters}\:\mathrm{C}\:\mathrm{are}\:\mathrm{next}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other}.\:\mathrm{to} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{where}\:\mathrm{two} \\ $$$$\mathrm{C}\:\mathrm{are}\:\mathrm{next}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other},\:\mathrm{we}\:\mathrm{can}\: \\ $$$$\mathrm{proceed}\:\mathrm{as}\:\mathrm{above},\:\mathrm{but}\:\mathrm{treat}\:\mathrm{CC}\:\mathrm{as}\:\mathrm{a} \\ $$$$\mathrm{single}\:\mathrm{letter}.\:\mathrm{that}\:\mathrm{means}\: \\ $$$$\mathrm{YXYXYXY} \\ $$$$\Rightarrow\mathrm{3}!×\mathrm{C}_{\mathrm{3}} ^{\mathrm{4}} \\ $$$$ \\ $$$$\Rightarrow\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{such}\:\mathrm{that}\:\mathrm{neither} \\ $$$$\mathrm{two}\:\mathrm{S}\:\mathrm{nor}\:\mathrm{two}\:\mathrm{C}\:\mathrm{are}\:\mathrm{next}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other}\:\mathrm{is}: \\ $$$$\mathrm{4}!/\mathrm{2}!×\mathrm{C}_{\mathrm{3}} ^{\mathrm{5}} −\mathrm{3}!×\mathrm{C}_{\mathrm{3}} ^{\mathrm{4}} =\mathrm{24}/\mathrm{2}×\mathrm{10}−\mathrm{6}×\mathrm{4}=\mathrm{96} \\ $$
Commented by Tinkutara last updated on 07/Oct/17
Thank you very much Sir!
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$
Commented by mrW1 last updated on 07/Oct/17
I′m interessed what is the solution in your book.
$$\mathrm{I}'\mathrm{m}\:\mathrm{interessed}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{in} \\ $$$$\mathrm{your}\:\mathrm{book}. \\ $$
Commented by Tinkutara last updated on 07/Oct/17
It does not explain much.
$$\mathrm{It}\:\mathrm{does}\:\mathrm{not}\:\mathrm{explain}\:\mathrm{much}. \\ $$

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