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If-x-y-1-then-r-0-n-r-n-C-r-x-r-y-n-r-equals-




Question Number 63943 by gunawan last updated on 11/Jul/19
If   x+y=1, then  Σ_(r=0) ^n  r ^n C_r  x^r y^(n−r)  equals
Ifx+y=1,thennr=0rnCrxrynrequals
Commented by mathmax by abdo last updated on 11/Jul/19
x+y =1 ⇒(x+y)^n  =1 ⇒Σ_(r=0) ^n  C_n ^r  x^r  y^(n−r)   =1  ⇒  y^n  +Σ_(r=1) ^n  C_n ^r  x^r  y^(n−r)  =1  let derivate (y fixed) ⇒  Σ_(r=1) ^n  C_n ^r  r x^(r−1)  y^(n−r)  =0 ⇒(1/x)Σ_(r=1) ^n  C_n ^r  x^r  y^(n−r ) =0   (x≠0) ⇒  Σ_(r=0) ^n  C_n ^r  x^r  y^(n−r)  −y^n   =0 ⇒Σ_(r=0) ^n  C_n ^r  x^r  y^(n−r)  =y^n   if x=0 ⇒y=1 and Σ_(r=0) ^n (...) =0
x+y=1(x+y)n=1r=0nCnrxrynr=1yn+r=1nCnrxrynr=1letderivate(yfixed)r=1nCnrrxr1ynr=01xr=1nCnrxrynr=0(x0)r=0nCnrxrynryn=0r=0nCnrxrynr=ynifx=0y=1andr=0n()=0
Answered by mr W last updated on 11/Jul/19
(x+y)^n =1  Σ_(r=0) ^n C_r ^n x^r y^(n−r) =1  Σ_(r=0) ^n rC_r ^n x^(r−1) y^(n−r) =0  Σ_(r=0) ^n rC_r ^n x^r y^(n−r) =0×x  ⇒Σ_(r=0) ^n rC_r ^n x^r y^(n−r) =0
(x+y)n=1nr=0Crnxrynr=1nr=0rCrnxr1ynr=0nr=0rCrnxrynr=0×xnr=0rCrnxrynr=0

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