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Q-what-is-the-greatest-coefficient-of-the-following-expansion-P-a-b-c-d-10-




Question Number 173531 by mnjuly1970 last updated on 13/Jul/22
      Q:    what is the greatest coefficient    of the following  expansion.         P:= ( a+ b +c +d )^( 10)          −−−−−−−
Q:whatisthegreatestcoefficientofthefollowingexpansion.P:=(a+b+c+d)10
Answered by mr W last updated on 13/Jul/22
 coefficient =((10!)/(m!n!h!k!))  with 0≤m,n,h,k≤10 and m+n+h+k=10  due to symmetry, largest coefficient  is when m,n,h,k are at most equal to  each other. two cases shoulde be checked:  m=n=h=3, k=1 or  m=n=3, k=h=2  (3!)^3 ×1!=216  (3!)^2 ×(2!)^2 =144 <216  ⇒largest coef.=((10!)/(144))=25200 ✓
coefficient=10!m!n!h!k!with0m,n,h,k10andm+n+h+k=10duetosymmetry,largestcoefficientiswhenm,n,h,kareatmostequaltoeachother.twocasesshouldebechecked:m=n=h=3,k=1orm=n=3,k=h=2(3!)3×1!=216(3!)2×(2!)2=144<216largestcoef.=10!144=25200
Commented by mnjuly1970 last updated on 13/Jul/22
thanks alot sir W
thanksalotsirW
Commented by Tawa11 last updated on 13/Jul/22
Great sir
Greatsir

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