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Category: Permutation and Combination

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Question Number 199054 by cortano12 last updated on 27/Oct/23 $$\:\:\boldsymbol{{x}} \\ $$ Answered by AST last updated on 27/Oct/23 $$\left(\mathrm{1}+{x}\right)\left(\mathrm{2}+{x}^{\mathrm{2}} \right)\left(\mathrm{3}+{x}^{\mathrm{3}} \right)\left(\mathrm{4}+{x}^{\mathrm{4}} \right)\left(\mathrm{5}+{x}^{\mathrm{5}} \right)\left(\mathrm{6}+{x}^{\mathrm{6}} \right)…

Given-the-number-of-consisting-of-4-digits-abcd-such-that-a-b-c-d-is-A-495-B-385-C-275-D-165-E-55-

Question Number 198283 by cortano12 last updated on 16/Oct/23 $$\mathrm{Given}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\:\mathrm{consisting} \\ $$$$\:\mathrm{of}\:\mathrm{4}\:\mathrm{digits}\:\mathrm{abcd}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\mathrm{a}\leqslant\mathrm{b}\leqslant\mathrm{c}\leqslant\mathrm{d}\:\mathrm{is}\:… \\ $$$$\:\left(\mathrm{A}\right)\:\mathrm{495}\:\:\:\left(\mathrm{B}\right)\:\mathrm{385}\:\:\:\:\:\left(\mathrm{C}\right)\:\mathrm{275} \\ $$$$\:\left(\mathrm{D}\right)\:\mathrm{165}\:\:\:\:\left(\mathrm{E}\right)\:\mathrm{55}\: \\ $$ Commented by mr W last…

Question-198231

Question Number 198231 by pticantor last updated on 14/Oct/23 Answered by witcher3 last updated on 14/Oct/23 $$\mathrm{C}_{\mathrm{n}} ^{\mathrm{p}} =\frac{\mathrm{n}!}{\mathrm{p}!.\left(\mathrm{n}−\mathrm{p}\right)!} \\ $$$$\mathrm{pC}_{\mathrm{n}} ^{\mathrm{p}} =\frac{\mathrm{n}!}{\left(\mathrm{p}−\mathrm{1}\right)!.\left(\mathrm{n}−\mathrm{p}\right)!}=\mathrm{n}.\frac{\left(\mathrm{n}−\mathrm{1}\right)!}{\left(\mathrm{p}−\mathrm{1}\right)!.\left(\mathrm{n}−\mathrm{1}−\left(\mathrm{p}−\mathrm{1}\right)\right)!} \\ $$$$=\mathrm{nC}_{\mathrm{n}−\mathrm{1}}…