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Question
Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?
Solution
4-digit numbers are to be formed using the digits, 1, 2, 3, 4, and 5.
There will be as many 4-digit numbers as there are permutations of 5 different digits taken 4 at a time
Therefore, required number of 4 digit numbers =
5P4 = `(5!)/((5 - 4)!) = (5!)/(1!)`
= 1x 2 x 3 x 4 x 5 = 120
Among the 4-digit numbers formed by using the digits, 1, 2, 3, 4, 5, even numbers end with either 2 or 4.
The number of ways in which units are filled with digits is 2.
Since the digits are not repeated and the units place is already occupied with a digit (which is even), the remaining places are to be filled by the remaining 4 digits.
Therefore, the number of ways in which the remaining places can be filled is the permutation of 4 different digits taken 3 at a time.
Number of ways of filling the remaining places =
4P3 = `(4!)/((4 - 3)!) = (4!)/(1!)`
= 4 × 3 × 2 × 1 = 24
Thus, by multiplication principle, the required number of even numbers is
24 × 2 = 48
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