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Question Number 220877 by Spillover last updated on 20/May/25
Answered by Rasheed.Sindhi last updated on 20/May/25
$$\left({a}\right)\:\left({n}+\mathrm{2}\right)!+\left({n}+\mathrm{1}\right)!+{n}! \\ $$$$=\left({n}+\mathrm{2}\right)\left({n}+\mathrm{1}\right){n}!+\left({n}+\mathrm{1}\right){n}!+{n}! \\ $$$$={n}!\left(\left({n}+\mathrm{2}\right)\left({n}+\mathrm{1}\right)+\left({n}+\mathrm{1}\right)+\mathrm{1}\right) \\ $$$$\left.={n}!\left\{\:\left({n}+\mathrm{1}\right)\left({n}+\mathrm{3}\right)+\mathrm{1}\right)\right\} \\ $$$$={n}!\left({n}^{\mathrm{2}} +\mathrm{4}{n}+\mathrm{4}\right) \\ $$$$={n}!\left({n}+\mathrm{2}\right)^{\mathrm{2}} \\ $$
Commented by Spillover last updated on 21/May/25
$${correct} \\ $$
Answered by Rasheed.Sindhi last updated on 20/May/25
$$\left({b}\right)\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)!−\left({n}−\mathrm{2}\right)! \\ $$$$\:\:\:\:\:\:=\left({n}−\mathrm{2}\right)!\left({n}−\mathrm{1}−\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:=\left({n}−\mathrm{2}\right)!\left({n}−\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:=\left({n}−\mathrm{2}\right)^{\mathrm{2}} \left({n}−\mathrm{3}\right)! \\ $$
Commented by Spillover last updated on 21/May/25
$${correct} \\ $$
Answered by Rasheed.Sindhi last updated on 20/May/25
$$\left({c}\right)\:\mathrm{15}!+\mathrm{4}\left(\mathrm{14}!\right) \\ $$$$\:\:\:\:\:=\mathrm{15}.\mathrm{14}!+\mathrm{4}\left(\mathrm{14}!\right) \\ $$$$\:\:\:\:=\mathrm{14}!\left(\mathrm{15}+\mathrm{4}\right) \\ $$$$\:\:\:\:\:=\mathrm{19}\left(\mathrm{14}!\right) \\ $$
Answered by Rasheed.Sindhi last updated on 20/May/25
$$\left({d}\right)\:\mathrm{26}!−\mathrm{25}! \\ $$$$\:\:\:\:\:\:=\mathrm{26}\left(\mathrm{25}!\right)−\mathrm{25}! \\ $$$$\:\:\:\:\:\:=\mathrm{25}!\left(\mathrm{26}−\mathrm{1}\right) \\ $$$$\:\:\:\:\:=\mathrm{25}^{\mathrm{2}} \left(\mathrm{24}!\right) \\ $$
Commented by Spillover last updated on 21/May/25
$${correct} \\ $$