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#1 Re: Ganesh's Puzzles » Number Theory » 2009-04-30 19:08:07

M=10^5(m5)+10^4(m4)+10^3(m3)+10^2(m2)+10^1(m1)+10^0(m0) ≤ 10^5
⇒m5=0
⇒∑(10^i)mi ≤ 10^5 (i=1,2,3,4)...(1)
∑mi=10....(2) So the inequalility holds as max. value of M is 91000
∴No of M's satisfying (1)&(2) is = coefficient of x^10 in (x^0+x^1+x^2+.....upto ∞ terms )^5 - (5 c 4) (as mi can't take value 10;0≤mi≤10;0<x<1)
                                             =      ,,         ,,   ,,     ,, (1-x)^(-5) - (5 c 4)=(14 c 4) - 5=1001-5=996

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