`Example. How many triples of nonnegative integers (x1, x2, x3) satisfy the equation x1 + x2 + x3 = 7? (using Separation into types)`

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

We need to find the number of triples of nonnegative integers $(x_1, x_2, x_3)$ that satisfy the equation $x_1 + x_2 + x_3 = 7$. This is a classic problem of finding the number of nonnegative integer solutions to a linear equation, which can be solved using the "stars and bars" theorem

step 2

According to the "stars and bars" theorem, the number of ways to distribute $n$ indistinguishable objects (stars) into $k$ distinguishable bins (variables) is given by the binomial coefficient $\binom{n+k-1}{k-1}$

step 3

In our case, $n = 7$ (the total sum) and $k = 3$ (the number of variables). Therefore, we need to calculate $\binom{7+3-1}{3-1} = \binom{9}{2}$

step 4

The binomial coefficient $\binom{9}{2}$ is calculated as follows:
$\binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36$

Answer

36

Key Concept

Stars and Bars Theorem

Explanation

The stars and bars theorem is used to find the number of ways to distribute $n$ indistinguishable objects into $k$ distinguishable bins. In this problem, it helps us determine the number of nonnegative integer solutions to the equation $x_1 + x_2 + x_3 = 7$.

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