Question

Compute how many 7-digit numbers can be made from the digits 1,2,3,4,5,6,7 if there is no repetition and the odd digits must appear in an unbroken sequence

Answer 1

There are four odd digits, and the odd digits sequence begin on the 1st empty spot, 2nd, 3r or 4rth.
So basically we have four choices, where the first digit is odd.
We get
Four ways (4!) to arrange odd digits.
Three ways (3!) to arrange even digits.
Total arrangements = 4 x 4! x 3!
Simplify
Total arrangements= 4 x 4 x 3 x 2 x 1 x 3 x 2 x 1
Total arrangements= 576

Answer: 576