The number of integers, greater than $7000$ that can be formed, using the digits $3,5,6,7,8$ without repetition, is
- 168
- 120
- 220
- 48
The Correct Option is A
Approach Solution - 1
Four digit numbers greater than
Five digit number
Total number greater than
Approach Solution -2
To form numbers greater than 7000, consider two cases:
Case 1: Four-digit numbers greater than 7000
The thousands place must be 7 or 8 (2 choices). The remaining 3 digits can be arranged in:
\[ 4 \times 3 \times 2 \text{ ways}. \]
Thus, the total number of four-digit numbers greater than 7000 is:
\[ 2 \times 4 \times 3 \times 2 = 48. \]
Case 2: Five-digit numbers
All five-digit numbers formed using these digits are valid. The total number of arrangements for five digits is:
\[ 5! = 120. \]
Total numbers greater than 7000:
The total number of numbers greater than 7000 is:
\[ 48 + 120 = 168. \]
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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