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Binomial Chapter :: Help

How to solve this:

Find The value of :: \(\displaystyle{\sum _{ 0 }^{ n }{ { { ^{ n }{ P } } }_{ r } }} \).

here P(n,r) means permutation of n things taken r at a time .

I have not study Binaomial chapter in detaied Yet . But I need this sumition whlile solving an question of PNC .

PLease Help , I have no idea for such summitions , Please Explain , Thanks a lot

Note by Karan Shekhawat
2 years, 8 months ago

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We can actually evaluate the sum as \(\Gamma(n+1,1)\) , where \(\Gamma(a,x)\) is the Upper Incomplete Gamma function .

\[\begin{align} \sum_{r=0}^{n} \binom{n}{r} r! &= \sum_{r=0}^{n} \binom{n}{r} \int_{0}^{\infty} e^{-x} x^r \mathrm{d}x\\ &= \int_{0}^{\infty} \sum_{r=0}^{n} \binom{n}{r} e^{-x} x^r \mathrm{d}x\\ &= \int_{0}^{\infty} e^{-x} (1+x)^{n} \mathrm{d}x\\ &=\boxed{e \cdot \Gamma(n+1,1)}\end{align}\] .

This solution has been inspired from the genius Pratik Shastri's solution to this problem .

Azhaghu Roopesh M - 2 years, 8 months ago

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Do you know how to add a question in a white box?

U Z - 2 years, 8 months ago

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Add ">>" before the text.

Pratik Shastri - 2 years, 8 months ago

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Use \boxed{}

Azhaghu Roopesh M - 2 years, 8 months ago

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@Azhaghu Roopesh M No , I am asking the way Karan posted the question in a white box , I am toogling latex but it does'nt show the code for it

U Z - 2 years, 8 months ago

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@U Z Nope. Let's ask @KARAN SHEKHAWAT himself .

Azhaghu Roopesh M - 2 years, 8 months ago

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Ohh , so it is gamma summition . Can we use any other approach ? Since I didn't Know anything about gamma function , or can you please tell something about this gamma function ? I'am nwe to it . Thanks

Karan Shekhawat - 2 years, 8 months ago

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Well even I am new to Gamma function . If you study the topic from Wolfram Mathworld , you will get some understanding . It's actually quite easy , we are just exploiting the properties of Gamma Function .

Azhaghu Roopesh M - 2 years, 8 months ago

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Can you please explain the second step , not getting

U Z - 2 years, 8 months ago

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Well in the second step I've just taken the Summation into the integral .

\[\begin{align} \int_{0}^{\infty} \sum_{r=0}^{n} \binom{n}{r} e^{-x} x^r \mathrm{d}x\\ \end{align}\] and we all know that \[ \sum_{r=0}^{n} \binom{n}{r} x^r = (1+x)^{n} \] . And after that we just use the definition of Incomplete Upper Gamma Function .

Azhaghu Roopesh M - 2 years, 8 months ago

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@Azhaghu Roopesh M I am not getting how did you take the summation into integral , how is this valid , can you explain in more detail please

U Z - 2 years, 8 months ago

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@U Z The summation is dependant on x [i.e.dx] and the summation is independant of x [it's dependant on r]so I don't think that we should have a problem taking it into the integral .

Azhaghu Roopesh M - 2 years, 8 months ago

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@Azhaghu Roopesh M their upper limits are different , then also can we slide the summation into integral (how?) any reason?

U Z - 2 years, 8 months ago

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@U Z Check this out.

Pratik Shastri - 2 years, 8 months ago

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@U Z Since their limits are different, we can almost treat the Summation Term as a constant w.r.t the Integral (That's what I think) . BTW I'll ask Pratik Shastri and let you know

Azhaghu Roopesh M - 2 years, 8 months ago

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