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Question Number 26480 by NECx last updated on 26/Dec/17
$${Find}\:{the}\:{number}\:{of}\:{all}\:{possible} \\ $$$${different}\:{words}\:{into}\:{which}\:{the} \\ $$$${word}\:{INTERFERE}\:{can}\:{be}\: \\ $$$${converted}\:{by}\:{change}\:{of}\:{place}\:{of}\: \\ $$$${letters},{if}\:{no}\:{two}\:{consonants} \\ $$$${must}\:{be}\:{together}. \\ $$
Answered by mrW1 last updated on 26/Dec/17
$${The}\:{consonants}\:{NTRFR}\:{must}\:{be} \\ $$$${separated}\:{by}\:{the}\:{other}\:\mathrm{4}\:{letters}. \\ $$$${XYXYXYXYX} \\ $$$${Y}={position}\:{of}\:{vowels} \\ $$$${X}={position}\:{of}\:{the}\:{consonants} \\ $$$${To}\:{arrange}\:{the}\:\mathrm{4}\:{vowels}\:\left({IEEE}\right):\: \\ $$$$\mathrm{4}\:{ways} \\ $$$$ \\ $$$${To}\:{arrange}\:{the}\:\mathrm{5}\:{consonants}: \\ $$$$\frac{\mathrm{5}!}{\mathrm{2}!}=\mathrm{5}×\mathrm{4}×\mathrm{3}=\mathrm{60}\:{ways} \\ $$$$ \\ $$$${Number}\:{of}\:{words}: \\ $$$$\mathrm{4}×\mathrm{60}=\mathrm{240} \\ $$
Commented by NECx last updated on 26/Dec/17
$${thanks}\:{boss} \\ $$