# Thread: No. of solutions for x1 + x2 + x3 + x4 + x5 + x6 < 10

1. ## No. of solutions for x1 + x2 + x3 + x4 + x5 + x6 < 10

Find the number of non-negative integer solutions of the inequality
x1 + x2 + x3 + x4 + x5 + x6 < 10.

If this was -
Find the number of non-negative integer solutions of the equation
x1 + x2 + x3 + x4 + x5 + x6 = 10

I would view this as having 10 identical items to distribute among 6 people and would use $\binom{10+6-1}{10}$.

However Im not sure how to handle it with when it's an equality. Would I write it as a summation formula?

$\sum_{n=0}^9 {n+6-1\choose n}$

2. ## Re: No. of solutions for x1 + x2 + x3 + x4 + x5 + x6 < 10

"10 identical items..." - ??? What makes you think that they are identical?

Take a look at this:
number theory - Non-negative integral solutions of $X_1+X_2+X_3+X_4<n$ - Mathematics - Stack Exchange

3. ## Re: No. of solutions for x1 + x2 + x3 + x4 + x5 + x6 < 10

They are identical in that they I am viewing this as having ten 'ones' that are being distributed among 6 people....
Taking x1 + x2 + x3 + x4 + x5 + x6 = 10
So x1 could have 5 'ones' and x2 - x6 could have one 'one' each. Which would be a solution for the equation.

4. ## Re: No. of solutions for x1 + x2 + x3 + x4 + x5 + x6 < 10

Originally Posted by nukenuts
Find the number of non-negative integer solutions of the inequality
x1 + x2 + x3 + x4 + x5 + x6 < 10.
If this was -
Find the number of non-negative integer solutions of the equation
x1 + x2 + x3 + x4 + x5 + x6 = 10
I would view this as having 10 identical items to distribute among 6 people and would use $\binom{10+6-1}{10}$.
However Im not sure how to handle it with when it's an equality. Would I write it as a summation formula?
$\color{blue}\sum_{n=0}^9 {n+6-1\choose n}$
Your solution is absolutely correct. Way to go.

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# There are 3003 nonnegative integer solutions to the equation x1 x2 x3 x4 x5 x6=10. How many such solutions are there to the inequality x1 x2 x3 x4 x5 x6<10?

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