Task 401

The task

In 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 numbers from the set $1-15$ in the first column, 5 numbers from $16-30$ in the second column, 4 numbers $31-45$ in the third column (skipping the WILD square in the middle), 5 numbers from $46-60$ in the fourth column and 5 numbers from $61-75$ in the last column. One possible 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("$31$",(2.5,0.5)); label("$32$",(2.5,1.5)); label("Wild",(2.5,2.5)); label("$34$",(2.5,3.5)); label("$35$",(2.5,4.5)); label("$50$",(3.5,0.5)); label("$49$",(3.5,1.5)); label("$48$",(3.5,2.5)); label("$47$",(3.5,3.5)); label("$46$",(3.5,4.5)); label("$71$",(4.5,0.5)); label("$72$",(4.5,1.5)); label("$73$",(4.5,2.5)); label("$74$",(4.5,3.5)); label("$75$",(4.5,4.5)); [/asy] To play 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 diagonal going from top left to bottom right of a BINGO card, in order?