The number of permutations that can be made from the letters 'SUNDAY' without beginning with 'S' or without ending with 'Y' ??

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Number} \mathrm{of} \mathrm{words} \mathrm{without} \mathrm{beginning} \mathrm{with} \mathrm{S}\hspace{0.17em}\mathrm{and} \mathrm{without} \mathrm{ending} \mathrm{with} \mathrm{Y} \\ =\mathrm{Total} \mathrm{Words}-\left(\mathrm{Words} \mathrm{which} \mathrm{start} \mathrm{with} \mathrm{S}\right)-\left(\mathrm{Words} \mathrm{which} \mathrm{end} \mathrm{with} \mathrm{Y}\right) \\ +\left(\mathrm{Words} \mathrm{which} \mathrm{start} \mathrm{with} \mathrm{S} \mathrm{and} \mathrm{end} \mathrm{with} \mathrm{Y}\right) \\ \mathrm{S},\mathrm{U},\mathrm{N},\mathrm{D},\mathrm{A},\mathrm{Y}\to 6 \mathrm{letters} \\ \mathrm{Total} \mathrm{words}=6!=6\times 5\times 4\times 3\times 2\times 1=720 \\ \mathrm{Now} \\ \frac{1}{6} \mathrm{words} \mathrm{will} \mathrm{start} \mathrm{with} \mathrm{S} \\ \Rightarrow \mathrm{Words} \mathrm{which} \mathrm{start} \mathrm{with} \mathrm{S}=\frac{1}{6}\times 720=120 \\ \mathrm{Similarly} \\ \mathrm{Words} \mathrm{which} \mathrm{end} \mathrm{with} \mathrm{Y}=\frac{1}{6}\times 720=120 \\ \mathrm{For} \mathrm{words} \mathrm{which} \mathrm{start} \mathrm{with} \mathrm{S} \mathrm{and} \mathrm{end} \mathrm{with} \mathrm{Y}, \mathrm{first} \mathrm{and} \mathrm{last} \mathrm{place} \mathrm{is} \mathrm{fixed}, \mathrm{hence} \\ \mathrm{we} \mathrm{have} \mathrm{arrange} \mathrm{remaining} 4 \mathrm{letters} \mathrm{at} \mathrm{remaining} 4 \mathrm{places} \\ \Rightarrow \mathrm{Words} \mathrm{which} \mathrm{start} \mathrm{with} \mathrm{S} \mathrm{and} \mathrm{end} \mathrm{with} \mathrm{Y}=4!=4\times 3\times 2\times 1=24 \\ \Rightarrow \mathrm{Number} \mathrm{of} \mathrm{words} \mathrm{without} \mathrm{beginning} \mathrm{with} \mathrm{S}\hspace{0.17em}\mathrm{and} \mathrm{without} \mathrm{ending} \mathrm{with} \mathrm{Y} \\ =720-120-120+24 \\ \mathbf{=}\mathbf{504}\mathbf{ }\left(\mathbf{Answer}\right) \end{gathered}$$

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