N(A or B) = N(A) + N(B) - N(A and B) A = starts with a 3 B = ends with a 7 A and B = starts with a 3 and ends with a 7 N(A): Choose the 1st digit in 1 way, as a 3. Choose 2nd digit 9 ways Choose 3rd digit 8 ways Choose 4th digit 7 ways Choose 5th digit 6 ways Choose 6th digit 5 ways Choose 7th digit 4 ways Choose 8th digit 3 ways Choose 9th digit 2 ways Choose 10th digit 1 way. N(A) = 1*9*8*7*6*5*4*3*2*1 = 9! = 362880 N(B): Choose the 10th digit in 1 way, as a 7. Choose 1st digit 9 ways Choose 2nd digit 8 ways Choose 3rd digit 7 ways Choose 4th digit 6 ways Choose 5th digit 5 ways Choose 6th digit 4 ways Choose 7th digit 3 ways Choose 8th digit 2 ways Choose 9th digit 1 way. N(B) = 1*9*8*7*6*5*4*3*2*1 = 9! = 362880 N(A and B): Choose the 1st digit in 1 way, as a 3. Choose the 10th digit in 1 way, as a 7. Choose 2nd digit 8 ways Choose 3rd digit 7 ways Choose 4th digit 6 ways Choose 5th digit 5 ways Choose 6th digit 4 ways Choose 7th digit 3 ways Choose 8th digit 2 ways Choose 9th digit 1 way. N(A and B) = 1*1*8*7*6*5*4*3*2*1 = 8! = 40320. N(A or B) = N(A) + N(B) - N(A and B) N(A or B) = 362880 + 362880 - 40320 = 685440 Or you can write it 2×9!-8! Or you can factor out 8! and write it as 8!(2×9-1) = 8!(18-1) = 8!×17 or 17×8!. All those equal 685440. Edwin