Task 1294

The task

In SHORT BINGO, a $5\times5$ card is filled by marking the middle square as WILD and placing 24 other numbers in the remaining 24 squares. Specifically a card is made by placing 5 distinct numbers from the set $1-10$ in the first column, 5 distinct numbers from $11-20$ in the second column, 4 distinct numbers $21-30$ in the third column (skipping the WILD square in the middle), 5 distinct numbers from $31-40$ in the fourth column and 5 distinct numbers from $41-50$ in the last column. One possible SHORT BINGO card is: [asy] for (int i=0; i<6;++i) { draw((i,0)--(i,5)); draw((0,i)--(5,i)); } label("$1$",(.5,0.5)); label("$2$",(.5,1.5)); label("$3$",(.5,2.5)); label("$4$",(.5,3.5)); label("$5$",(.5,4.5)); label("$20$",(1.5,0.5)); label("$19$",(1.5,1.5)); label("$18$",(1.5,2.5)); label("$17$",(1.5,3.5)); label("$16$",(1.5,4.5)); label("$21$",(2.5,0.5)); label("$22$",(2.5,1.5)); label("Wild",(2.5,2.5)); label("$24$",(2.5,3.5)); label("$25$",(2.5,4.5)); label("$40$",(3.5,0.5)); label("$39$",(3.5,1.5)); label("$38$",(3.5,2.5)); label("$37$",(3.5,3.5)); label("$36$",(3.5,4.5)); label("$41$",(4.5,0.5)); label("$42$",(4.5,1.5)); label("$43$",(4.5,2.5)); label("$44$",(4.5,3.5)); label("$45$",(4.5,4.5)); [/asy] To play SHORT BINGO, someone names numbers, chosen at random, and players mark those numbers on their cards. A player wins when he marks 5 in a row, horizontally, vertically, or diagonally. How many distinct possibilities are there for the values in the first column of a SHORT BINGO card? (The placement on the card matters, so the order of the numbers matters, so $5~4~3~2~1$ is to be considered different from $1~2~3~4~5$, for instance.)