The number of different solutions of the equation x+y+z=12, where each of x, y and z is a positive integer, is

Explanation

Using the stars and bars method for strictly positive integers, the number of solutions to x_1 + x_2 + \dots + x_r = n is given by \binom{n-1}{r-1}. Here, n=12, r=3, so \binom{12-1}{3-1} = \binom{11}{2} = \frac{11 \times 10}{2} = 55.

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