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Question Number 195674 by Rodier97 last updated on 07/Aug/23

$$\:\:\:\int_{\mathrm{0}} ^{\mathrm{4}} \:\frac{{x}!}{\mathrm{5}!\left({x}−\mathrm{5}\right)!}\:{dx}\:=\:??? \\$$

Commented by mr W last updated on 07/Aug/23

$${what}\:{do}\:{you}\:{mean}\:{with}\:\begin{pmatrix}{{x}}\\{\mathrm{5}}\end{pmatrix}\:? \\$$

Commented by Rodier97 last updated on 07/Aug/23

$$\\$$$$\:\:\:\:{C}_{{x}} ^{\:\mathrm{5}} =\frac{{x}!}{\mathrm{5}!\left({x}−\mathrm{5}\right)!} \\$$

Commented by mr W last updated on 07/Aug/23

$${in}\:{definition}\:\:\begin{pmatrix}{{x}}\\{\mathrm{5}}\end{pmatrix}={C}_{{x}} ^{\:\mathrm{5}} =\frac{{x}!}{\mathrm{5}!\left({x}−\mathrm{5}\right)!}\: \\$$$${we}\:{have}\:{x}\in{N}\:{and}\:{x}\geqslant\mathrm{5}. \\$$$${but}\:{in}\:\int_{\mathrm{0}} ^{\mathrm{4}} {f}\left({x}\right){dx}\:{we}\:{have}\: \\$$$${x}\in{R}\:{and}\:\mathrm{0}\leqslant{x}\leqslant\mathrm{4}. \\$$

Commented by kapoorshah last updated on 07/Aug/23

$$\int_{\mathrm{0}} ^{\mathrm{4}} \frac{{x}\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}−\mathrm{3}\right)\left({x}−\mathrm{4}\right)}{\mathrm{120}}{dx} \\$$