N>40000, where N is divisible by 5. How many such 5 digit numbers can be formed using 0,1,3,5,7,9 without repetition.
Approach Solution - 1
The correct answer is 120.
Approach Solution -2
We are to form 5-digit numbers using the digits {0, 1, 3, 5, 7, 9} without repetition, under two conditions:
- The number must be greater than 40,000.
- The number is divisible by 5 (i.e. its last digit must be 0 or 5)
Case 1: Last Digit is 0
With the last digit fixed as 0, the first digit cannot be 0 and must make the number > 40,000. The available digits for the first position (excluding 0) are {1, 3, 5, 7, 9}. Since the number must be > 40,000, the first digit must be at least 4. Among our choices, only 5, 7, and 9 qualify.
Number of choices for the first digit: 3.
After fixing the first and last digits, there remain 4 digits for the three middle positions. The number of ways to fill these positions is:
P(4, 3) = 4 × 3 × 2 = 24
Total numbers for Case 1: 3 × 24 = 72.
Case 2: Last Digit is 5
With the last digit fixed as 5, the first digit must be chosen from the remaining digits {0, 1, 3, 7, 9} (0 is not allowed in the first position) and must be at least 4 to ensure the number is > 40,000. This leaves only 7 and 9.
Number of choices for the first digit: 2.
The remaining three positions (the 2nd, 3rd, and 4th digits) can be filled from the 4 remaining digits (from a total of 6, after fixing the first and last digits) in:
P(4, 3) = 4 × 3 × 2 = 24
Total numbers for Case 2: 2 × 24 = 48.
Total Count
Total 5-digit numbers = 72 + 48 = 120.
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Concepts Used:
Permutations
A permutation is an arrangement of multiple objects in a particular order taken a few or all at a time. The formula for permutation is as follows:
\(^nP_r = \frac{n!}{(n-r)!}\)
nPr = permutation
n = total number of objects
r = number of objects selected
Types of Permutation
- Permutation of n different things where repeating is not allowed
- Permutation of n different things where repeating is allowed
- Permutation of similar kinds or duplicate objects