The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1,3,5,7,9 without repetition, is
- 72
120
6
12
The Correct Option is A
Approach Solution - 1
We are tasked with finding the number of four-digit numbers that satisfy the following conditions:
- The number must be between 5000 and 10000.
- The digits must come from the set \( \{1, 3, 5, 7, 9\} \).
- No digit may be repeated.
The first digit must ensure the number is between 5000 and 10000. Therefore, it must be \( 5 \), \( 7 \), or \( 9 \). This gives us 3 choices for the first digit.
Once the first digit is chosen, there are 4 remaining digits in the set \( \{1, 3, 5, 7, 9\} \) (excluding the chosen first digit). The second digit can then be any one of these 4 digits, giving 4 choices for the second digit. For the third digit, we are left with 3 choices (the remaining digits after the first two have been selected). Similarly, for the fourth digit, there are 2 remaining choices.
The total number of valid numbers is given by the product of choices for each position:
\[ 3 \times 4 \times 3 \times 2 = 72 \]
Thus, there are 72 four-digit numbers that satisfy the given conditions.
Approach Solution -2
The correct answer is (A) : 72
Numbers between 5000&10000
Using digits 1,3,5,7,9
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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
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