Given the word $S,T,P,M,9,5,4,3$, find the number of permutations where the numbers are totally separated.
First I use $4p4 \times 4p4$, where $4p4 = 24$. Then I considered the arrangements of numbers and letters.
$9,S,5,T,4,P,3,M$
$9,ST,5,P,4,M,3$
$9,S,5,TP,4,M,3$
$9,S,5,T,4,PM,3$
$S,9,T,5,P,4,M,3$
After including the conditions above I found out $4p4 \times 4p4 \times 5$ which is $24 \times 24 \times 5 =2880$. But the answer should be $8640$. Where did I do wrong?
I agree with you. There are five orders of numbers and letters that have no numbers next to each other, then $4!$ ways to order the numbers and $4!$ ways to order the letters.