Let $n$ be an odd nine digit number divisible by $375$. How many $n$ exist such that the second to second-last digit are in an increasing order. So if $n=\overline {a_1a_2a_3a_4...a_9}$
$a_2\le a_3\le a_4\le a_5\le...\le a8$
$n=2k+1$
$n=375q$
I'm completely stumped. There's so many cases to look at. I don't know how to incorporate the criteria of being divisible by $375$ with the criteria of the increasing order of the  middle digits. I can find either of them, but I don't know how to combine them?
 A: The final three digits must form a number divisible by 125, and with the added requirement that for such a number $a,b,c$ we need $a \le b$ (the $c$ being the units digit of our number, so not affected) there are only the ending three digits 000, 125, 250, 375. The other multiples 500, 625, 750,875 all break the nondecreasing digit rule. Oops, yuu also need odd numbers, so only 125 and 375 qualify for endings.
For each of these it should then be relatively easy to find how many ways to pad it out to a nondecreasing sequence with sum 0 mod 3. For example with last digits 375 we need a decreasing sequence of length 6 all at least 3, and we have a final leading digit which has no requirement re. decreasing with the other digits. 
We're in luck in a sense, since if the number is say hvwxyz375 then the digit sum without the h determines the high digit h mod 3, and also h cannot be 0 in a 9 digit number, so there are 3 choices each for the digit h, once it is known mod 3. For hvwxyz375 we can fill out the subsequence vwxyz using any digits from 0,1,2,3. The nondecreasing requirement then means we have for this subsequence say a 0,s, then b 1's, then c 2's, then d 3's where each of a,b,c,d is arbitrary nonnegative and a+b+c+d=5. The number of solutions to that is, using stars and bars, $\binom{5+3}{3}=56.$ Multiplying that by 3 for the choice of h then gives 168 values of $n$ ending with 375. A similar analysis for strings ending 125 gives another 18 values of $n$ So (if this is right) the total number of $n$ is $168+18=186.$
