Find how many different odd 4-digit numbers less than
4000 can be made from the digits 1,2,3,4,5,6 and 7 if
no digit may be repeated.
We must break this into two cases:
Case 1: The first digit is 2.
Case 2: The first digit is 1 or 3.
Case 1: The first digit is 2
We choose the first digit 1 way (as 2).
We can choose the fourth digit 3 ways, as 1,3,5, or 7.
We can choose the second digit any of the 5 remaining digits.
We can choose the third digit any of the 4 remaining digits.
That's 1*4*5*4 = 80 ways
Case 2: The first digit is 1 or 3.
We choose the first digit 2 ways (as 1 or 3)
We choose the fourth digit 3 ways (5,7, or whichever of 1 and 3 that
wasn't chosen as the first digit.
We can choose the second digit any of the 5 remaining digits.
We can choose the third digit any of the 4 remaining digits.
That's 2*3*5*4 = 120 ways
Total: 80+20 = 200
Edwin