how many 7-digit telephone numbers are possible if the first digit cannot be zero and
A) every other digit is odd
Case 1: 1st, 3rd, 5th, 7th digits are odd
Choose the 1st digit 5 ways (1,3,5,7,9}
Choose the 2nd digit 10 ways {0,1,2,3,4,5,6,7,8,9}
Choose the 3rd digit 5 ways {1,3,5,7,9}
Choose the 4th digit 10 ways {0,1,2,3,4,5,6,7,8,9}
Choose the 5th digit 5 ways {1,3,5,7,9}
Choose the 6th digit 10 ways {0,1,2,3,4,5,6,7,8,9}
Choose the 7th digit 5 ways {1,3,5,7,9}
That's 5·10·5·10·5·10·5 = (5^4)(10^3) = 625000 for case 1
Case 2: 2nd, 4th, 6th digits are odd
Choose the 1st digit 9 ways (1,2,3,4,5,6,7,8,9}
Choose the 2nd digit 5 ways {1,3,5,7,9}
Choose the 3rd digit 10 ways {0,1,2,3,4,5,6,7,8,9}
Choose the 4th digit 5 ways {1,3,5,7,9}
Choose the 5th digit 10 ways {0,1,2,3,4,5,6,7,8,9}
Choose the 6th digit 5 ways {1,3,5,7,9}
Choose the 7th digit 10 ways {0,1,2,3,4,5,6,7,8,9}
That's 9·5·10·5·10·5·10 = 9(5^3)(10^3) = 1125000 for case 2
Total = 625000 + 1125000 = 1750000
B)the telephone number must start with 226 and end with 5
226---5
We only need to choose the 4th, 5th and 6th digits.
Choose the 4th digit 10 ways {0,1,2,3,4,5,6,7,8,9}
Choose the 5th digit 10 ways {0,1,2,3,4,5,6,7,8,9}
Choose the 6th digit 10 ways {0,1,2,3,4,5,6,7,8,9}
Answer: 10·10·10 = 10^3 = 1000
C) no repetitions are allowed
Choose the 1st digit as any of the 9 unused digits.
(1,2,3,4,5,6,7,8,9} (can't be 0)
Choose the 2nd digit as any of the 9 unused digits. (can be 0)
Choose the 3rd digit as any of the 8 unused digits.
Choose the 4th digit as any of the 7 unused digits.
Choose the 5th digit as any of the 6 unused digits.
Choose the 6th digit as any of the 5 unused digits.
Choose the 7th digit as any of the 4 unused digits.
Answer 9·9·8·7·6·5·4 = 544320
Edwin
