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  4. The number of positive integral solutions of the inequality x+y+z≤ 20 is

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Q. The number of positive integral solutions of the inequality $x+y+z\leq 20$ is

NTA AbhyasNTA Abhyas 2020Permutations and Combinations

Solution:

The number of solutions (positive integer) of $x+y+z=r$ is $^{r-1} C_{2}$
$\therefore $ The desired number of solutions is $\sum_{r=3}^{20} r^{r-1} C_{2}$
$={ }^{2} C_{2}+{ }^{3} C_{2}+{ }^{4} C_{2}+\ldots \ldots .+{ }^{19} C_{2}$
$=\left({ }^{3} C_{3}+{ }^{3} C_{2}\right)+{ }^{4} C_{2}+\ldots \ldots+{ }^{19} C_{2}$
$={ }^{4} C_{3}+{ }^{4} C_{2}+\ldots \ldots .+{ }^{18} C_{2}+{ }^{19} C_{2}$
$={ }^{20} C_{3}$ (Adding in the same way) $=\frac{20 \times 19 \times 183}{3 \times 2}=1140$