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June 2014 p62 q5
2719
Find how many different numbers can be made from some or all of the digits of the number 1 345 789 if
all seven digits are used, the odd digits are all together and no digits are repeated,
the numbers made are even numbers between 3000 and 5000, and no digits are repeated,
the numbers made are multiples of 5 which are less than 1000, and digits can be repeated.
Solution
(i) To keep the odd digits together, treat them as a single unit. The odd digits are 1, 3, 5, 7, 9. Arrange these 5 digits in a block: 5! ways. The remaining two digits (4 and 8) can be arranged with this block in 3! ways. Thus, the total number of arrangements is:
\(5! \times 3! = 120 \times 6 = 720\)
(ii) For even numbers between 3000 and 5000, the first digit must be 3 or 4, and the last digit must be 4 or 8. Consider two cases:
Case 1: Start with 3 and end with 4. The middle two digits can be any of the remaining 5 digits. Thus, there are:
\(5 \times 4 = 20\)
Case 2: Start with 3 and end with 8. The middle two digits can be any of the remaining 5 digits. Thus, there are:
\(5 \times 4 = 20\)
Case 3: Start with 4 and end with 8. The middle two digits can be any of the remaining 5 digits. Thus, there are:
\(5 \times 4 = 20\)
Adding these cases gives:
\(20 + 20 + 20 = 60\)
(iii) For multiples of 5 less than 1000, the number must end in 5. Consider numbers with 1, 2, or 3 digits:
1-digit: Only 5 is possible.
2-digit: The first digit can be any of 1, 3, 4, 7, 8, 9 (6 choices).
3-digit: The first two digits can be any combination of the remaining 6 digits (6 choices for the first, 5 for the second).
