- #1
Woolyabyss
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Homework Statement
How many non-negative integer solutions are there to the equation
x1 + x2 + x3 + x4 + x5 < 11,
(i)if there are no restrictions?
(ii)How many solutions are there if x1 > 3?
(iii)How many solutions are there if each xi < 3?
Homework Equations
N/A
The Attempt at a Solution
(i) inequality equivalent to equality x1 + x2 + x3 + x4 + x5 + x6 = 10
(n+k-1)choose(k-1) = (10+6-1)choose(6-1) = 3003
(ii) if x1 > 3 ------------> x2 + x3 + x4 + x5 < 8
equality equivalent x2 + x3 + x4 + x5 +x6 = 7
again (7+5-1)choose(4) = 330
(iii)
Its this part I'm not certain about
like in (ii) let x1 instead be > 3 we find 7+5-1)choose(4) = 330
now let x1 and x2 be > 3
we find x3 + x4 + x5 < 5
----> x3 + x4 + x5 +x6 = 4
(4+4-1)choose(4-1) =35
finally let x1,x2 and x3 >3
then x4 +x5 <2
equality x4 + x5 + x6 =1
(1+3-1)choose(2) = 3
answer: 3003 - 35 - 330 - 3 = 2635
any help would be appreciated.