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Question

How many 3-digit even numbers can be made using the digits 1,2,3,4,6,7 if no digit is repeated?

Solution
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A three digit even number is to be formed from given 6 digits 1,2,3,4,6,7.
HTO
Since, for the number is to be even , so ones place can be filled by 2,4 or 6. So, there are 3 ways to fill ones place.
Since, repetition is not allowed , so tens place can be filled by remaining 5 digits. So, tens place can be filled in 5 ways.
Similarly, hundred's place can be filled by remaining 4 digits. So, hundred's place can be filled in 4 ways.
So,required number of ways in which three digit even numbers can be formed from the given digits is 4×5×3=60
Alternative Method:
3-digit even numbers are to be formed using the given six digits, ,2,3,4,6 and 7, without repeating the digits.
Then, units digits can be filled in 3 ways by any of the digits, 2,4 or 6.
Since the digits cannot be repeated in the 3-digit numbers and units place is already occupied with a digit (which is even), the hundreds and tens place is to be filled by the remaining 5 digits.
Therefore, the number of ways in which hundreds and tens place can be filled with the remaining 5 digits Is the permutation of 5 different digits taken 2 at a time.
5P2=5!(52)!=5!3!
Number of ways of filling hundreds and tens place
=5×4×3!3!=20
Thus, by multiplication principle, the required number of 3-digit numbers is 3×20=60

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