How many solutions are there to the equation x₁ + x₂ + x3 = 22, if each x, is a positive integer? Use the method shown in Example 9.6.5 to answer this question. Start by thinking of the number 22 as divided into 22 individual items and the variables x₁, x₂, and x3 as three categories into which these items are placed. Since each x, is a positive ✓✓integer, start by placing one item in each category and distribute the remaining items among the categories. The number of solutions to the equation that satisfy the given condition is the number f ways to place all the items into the categories. Thus, the answer is 231 x.

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