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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∻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 $x_1 + x_2 + x_3 = 7$.⚹◊How can we express the number of solutions to $x1 + x2 + x3 = 7$ using combinations and permutations?⍭ Generate me a similar question◊

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