NUMBER TUMBLE

Relevant wiki: Generating Functions

WLOG let $a <b< c < d$ Let $b = a + \alpha , c = a + \alpha + \beta , d = a + \alpha + \beta + \gamma$ for some $\alpha , \beta , \gamma \geq 1$

Now, we have to find positive integral solutions to the equation $a + b + c + d = 20$

Thus , we have to find number of solutions for

$4a +3\alpha + 2\beta + \gamma = 20$ . where $a, \alpha,\beta, \gamma \geq 1$

This comes out to be $\boxed{23 }$ .

Now, we can permute $a,b,c,d$ in $4! = 24$ ways .Thus the answer is $\color{#D61F06}{\boxed{23×24 = 552}}$