Counting regular occurrences in a specified interval

Sorry, didn't include an attachment earlier. I added an example file so this made more sense.

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This works perfectly. Thanks for your help Haseeb!

Sorry Haseeb, I spoke too soon on this being solved. I tried "stress testing" this using different intervals and payment dates, and it appears what the COUPNUM formula is doing is just counting the number of starting points for periods of a given length between the two interval end points. Here's a couple of examples.

Start Date: 7/10/2012
End Date: 10/11/2012
First Payment Date: 6/15/2010
Next Payment Date: 9/15/2012
Payment Frequency: 4 (quarterly)

This information tells me that my payments are 3/15, 6/15, 9/15 and 12/15 every year. So I'm basically trying to count the occurrences of these dates between 7/10/2012 and 10/11/2012. The answer here should be 1 occurrence of a payment (9/15/2012), but the COUPNUM function returns 2 because it's saying "7/10 starts a quarter, and 10/10 starts another quarter".

The same thing happens with other frequencies:

Start Date: 7/10/2012
End Date: 1/11/2013
First Payment Date: 6/15/2010
Next Payment Date: 12/15/2012
Payment Frequency: 2 (semi-annual)

Here, my payments are each 6/15 and 12/15, so 1 should occur during the specified interval (12/15/2012), but COUPNUM counts 2.

I can't just tack a "subtract 1" onto the formula across the board either because sometimes the count happens to be right. What I really need is the ability to recognize what date(s) payments happen each year based on the First Payment Date, Next Payment Date, and Frequency, and then to pick out all occurrences of those dates in the specified interval.

I think the first option has an extra parentheses at the end, but after removing that it looks like both versions work. I can't find an interval/frequency combo that doesn't match expected tally which is great. Marking this solved (for real this time).

Sort of a followup and I don't know yet if I actually need to go this route, but if I wanted to only make one endpoint inclusive (e.g. if start date falls on a payment date don't count it, but if end date falls on a payment date do count it) how could I tweak? I assume I'd need a conditional "-1" somewhere. Probably could accomplish this with a clunky "IF" statement around your formulas, but you probably know of a cleaner way.

Really appreciate your help daddylonglegs. Thanks!