IntegerSolution Intermediate

Problem - 4276

Find the number of ordered quadruples of integer $(a, b, c, d)$ satisfying $1\le a < b < c < d \le 10$.


Let $x_1=a$, $x_2=b-a$, $x_3=c-b$, $x_4=d-c$, and $x_5=11-d$. Then we have $$x_1 + x_2+x_3+x_4+x_5=11$$

Meanwhile, all $x_i$, $(i=1, 2, 3, 4, 5)$ are positive integers. Therefore the desired answer is the number of positive integer solutions to the above equation which is $$\binom{11-1}{5-1}=\boxed{210}$$

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