Using the digits 3, 4, 5, 6, 7, 8, and 9, how many 7-digit numbers can be constructed if the number must begin with an odd digit and digits may not be repeated?

Well, there are 4 odd digits, so that's only 4 choices for the first digit. 

After that, you have six digits remaining, from which you must choose 4, 6 choose 4 = 15. And there are 4! = 24 ways to arrange those 4 digits. 

Altogether the number of possibilities is: 
4 * 15 * 24 = 1440