Question Number 184030 by Mastermind last updated on 02/Jan/23
$$\mathrm{How}\:\mathrm{many}\:\mathrm{words}\:\mathrm{can}\:\mathrm{be}\:\mathrm{made}\: \\ $$$$\mathrm{from}\:\mathrm{5}\:\mathrm{letters}\:\mathrm{if} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{all}\:\mathrm{letters}\:\mathrm{are}\:\mathrm{different} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{2}\:\mathrm{letters}\:\mathrm{are}\:\mathrm{identical} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{all}\:\mathrm{letters}\:\mathrm{are}\:\mathrm{different}\:\mathrm{but}\:\mathrm{2} \\ $$$$\mathrm{partucular}\:\mathrm{letters}\:\mathrm{cannot}\:\mathrm{be} \\ $$$$\mathrm{adjacent}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{M}.\mathrm{m} \\ $$
Answered by mr W last updated on 02/Jan/23
$$\left({a}\right) \\ $$$${ABCDE} \\ $$$$\Rightarrow\mathrm{5}!=\mathrm{120}\:{words}\:{can}\:{be}\:{formed}. \\ $$$$\left({b}\right) \\ $$$${ABCXX} \\ $$$$\Rightarrow\frac{\mathrm{5}!}{\mathrm{2}!}=\mathrm{60}\:{words}\:{can}\:{be}\:{formed}. \\ $$$$\left({c}\right) \\ $$$${ACBDE} \\ $$$$\Rightarrow\mathrm{5}!−\mathrm{4}!×\mathrm{2}!=\mathrm{72}\:{words}\:{can}\:{be}\:{formed}. \\ $$$${or} \\ $$$$\Rightarrow\mathrm{6}×\mathrm{2}!×\mathrm{3}!=\mathrm{72} \\ $$