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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.