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Question Number 184030 by Mastermind last updated on 02/Jan/23

How many words can be made   from 5 letters if  (a) all letters are different  (b) 2 letters are identical  (c) all letters are different but 2  partucular letters cannot be  adjacent.      M.m

$$\mathrm{How}\:\mathrm{many}\:\mathrm{words}\:\mathrm{can}\:\mathrm{be}\:\mathrm{made}\: \\ $$$$\mathrm{from}\:\mathrm{5}\:\mathrm{letters}\:\mathrm{if} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{all}\:\mathrm{letters}\:\mathrm{are}\:\mathrm{different} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{2}\:\mathrm{letters}\:\mathrm{are}\:\mathrm{identical} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{all}\:\mathrm{letters}\:\mathrm{are}\:\mathrm{different}\:\mathrm{but}\:\mathrm{2} \\ $$$$\mathrm{partucular}\:\mathrm{letters}\:\mathrm{cannot}\:\mathrm{be} \\ $$$$\mathrm{adjacent}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{M}.\mathrm{m} \\ $$

Answered by mr W last updated on 02/Jan/23

(a)  ABCDE  ⇒5!=120 words can be formed.  (b)  ABCXX  ⇒((5!)/(2!))=60 words can be formed.  (c)  ACBDE  ⇒5!−4!×2!=72 words can be formed.  or  ⇒6×2!×3!=72

$$\left({a}\right) \\ $$$${ABCDE} \\ $$$$\Rightarrow\mathrm{5}!=\mathrm{120}\:{words}\:{can}\:{be}\:{formed}. \\ $$$$\left({b}\right) \\ $$$${ABCXX} \\ $$$$\Rightarrow\frac{\mathrm{5}!}{\mathrm{2}!}=\mathrm{60}\:{words}\:{can}\:{be}\:{formed}. \\ $$$$\left({c}\right) \\ $$$${ACBDE} \\ $$$$\Rightarrow\mathrm{5}!−\mathrm{4}!×\mathrm{2}!=\mathrm{72}\:{words}\:{can}\:{be}\:{formed}. \\ $$$${or} \\ $$$$\Rightarrow\mathrm{6}×\mathrm{2}!×\mathrm{3}!=\mathrm{72} \\ $$

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