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

   ∫_0 ^4  ((x!)/(5!(x−5)!)) dx = ???

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

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

what do you mean with  ((x),(5) ) ?

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

Commented by Rodier97 last updated on 07/Aug/23

      C_x ^( 5) =((x!)/(5!(x−5)!))

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

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

in definition   ((x),(5) )=C_x ^( 5) =((x!)/(5!(x−5)!))   we have x∈N and x≥5.  but in ∫_0 ^4 f(x)dx we have   x∈R and 0≤x≤4.

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

Commented by kapoorshah last updated on 07/Aug/23

∫_0 ^4 ((x(x−1)(x−2)(x−3)(x−4))/(120))dx

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

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