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?

There are four choices for the first digit (3, 5, 7, 9) and each of the choices can be followed by the other 6 digits in any order.

This will give 4 x 6! ways, or 4 x 6 x 5 x 4 x 3 x 2 x 1 ways.